Dynamic reconfiguration-switching of windings in an electric motor

ABSTRACT

Dynamic reconfiguration-switching of motor windings is optimized between winding-configurations. Acceleration is traded off in favor of higher velocity upon detecting an electric motor is at an optimal angular-velocity for switching to an optimal lower torque constant and voltage constant. The total back electromotive force (BEMF) is prohibited from inhibiting further acceleration to a higher angular-velocity.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation in part, and claims priority tocopending U.S. application Ser. No. 12/202,854, filed Sep. 2, 2008, nowPublished U.S. Application 2010/0052584A1, the entire contents of whichis incorporated herein by reference and is relied upon for claiming thebenefit of priority.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to the field of data tapetransport devices. The present invention specifically relates tooptimizing dynamic reconfiguration-switching of motor windings in anelectric motor.

2. Description of the Related Art

Magnetic tape provides a means for physically storing data. As anarchival medium, tape often comprises the only copy of the data. Tapemay be used to restore data lost in a disk-drive crash. A tape drive isused to store and retrieve data with respect to the magnetic tape. Anexample of a tape drive is the IBM TotalStorage Enterprise Tape Drive3592 manufactured by IBM Corporation. Tape drives are typically used incombination with an automated data storage library. For example, the IBMTotalStorage Enterprise Tape Library 3494 manufactured by IBMCorporation is an automated data storage library that may include one ormore tape drives and data storage media for storing data with respect tothe tape drives.

Tape drives frequently employ DC motors and feedback control systemswith motor drivers for operating the DC motors, to producewell-controlled motion parameters such as position, velocity, and tapetension. While the motors rotate, a back electromotive force (“BEMF”) isproduced by the tape drive electric motors. This BEMF voltage isproduced because the electric motors generate an opposing voltage whilerotating.

In tape drives such as the aforementioned IBM 3592 used in theEnterprise range, and the Linear Tape Open used in the mid-range, thecurrent tape linear velocity is limited by the tape reel'sangular-velocity. The tape reel's angular-velocity approaches a maximumwhen the BEMF produced by the reel motor approaches the voltage of thepower supply to the reel motors. Tape drives typically operate from +5and +12 V power supplies; therefore it is not possible to increase thepower supply voltage to increase the tape reel angular-velocity. Inlight of the foregoing, a need exists for a mechanism by which tape reelangular-velocity may be increased in tape transport systemsincorporating fixed-voltage power supplies.

SUMMARY OF THE INVENTION

While it is not possible to increase power supply voltage to increasetape reel angular-velocity in tape transport systems implementing afixed-voltage power supply, it is possible to decrease the BEMF in orderto increase the tape reel angular-velocity by reducing the torqueconstant and voltage constant of the reel motor. However, reducing thetorque constant of the reel motor decreases the tape reelangular-acceleration, thereby impacting performance. The presentinvention discloses apparatus and method embodiments of mechanisms toselectively either reduce the BEMF from the reel motor, thereforeincreasing the allowable tape reel angular-velocity for a fixed powersupply voltage, or maintain a higher tape reel angular-acceleration. Byuse of the following mechanism, either the tape linear velocity or tapelinear acceleration may be selectively increased relative to presentimplementations.

Accordingly, and in view of the foregoing, various exemplary method,system, and computer program product embodiments for dynamic and optimalreconfiguration-switching of motor windings are provided. In oneembodiment, by way of example only, dynamic reconfiguration-switching ofmotor windings is optimized between winding-configurations. Accelerationis traded off in favor of higher velocity upon detecting an electricmotor is at an optimal angular-velocity for switching to an optimallower torque constant and voltage constant. The total back electromotiveforce (BEMF) is prohibited from inhibiting further acceleration to ahigher angular-velocity.

In addition to the foregoing exemplary method embodiment, otherexemplary system and computer product embodiments are provided andsupply related advantages. The foregoing summary has been provided tointroduce a selection of concepts in a simplified form that are furtherdescribed below in the Detailed Description. This Summary is notintended to identify key features or essential features of the claimedsubject matter, nor is it intended to be used as an aid in determiningthe scope of the claimed subject matter. The claimed subject matter isnot limited to implementations that solve any or all disadvantages notedin the background.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the advantages of the invention will be readilyunderstood, a more particular description of the invention brieflydescribed above will be rendered by reference to specific embodimentsthat are illustrated in the appended drawings. Understanding that thesedrawings depict only typical embodiments of the invention and are nottherefore to be considered to be limiting of its scope, the inventionwill be described and explained with additional specificity and detailthrough the use of the accompanying drawings, in which:

FIG. 1 is a diagram illustrating the tape path in a tape transportsystem;

FIG. 2 is a block diagram of a motor control or driver circuit;

FIG. 3 is a portion of control circuit;

FIG. 4 is a first embodiment of motor coils with winding switches;

FIG. 5 is a second embodiment of motor coils with winding switches;

FIG. 6 is an exemplary flowchart for operation;

FIG. 7 is a table diagram illustrating an exemplary derivation of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings anddynamically switching between a 3-winding-configuration motor fortrading off angular-acceleration in favor of increased angular-velocitybetween each successive winding-configuration, with N=3;

FIG. 8 is a table diagram illustrating an exemplary profile of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings in a3-winding-configuration motor for trading off angular-acceleration infavor of increased angular-velocity between each successivewinding-configuration;

FIG. 9 is a table diagram illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 3winding-configurations;

FIG. 10 is a graph diagram illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 3winding-configurations;

FIG. 11 is a table diagram illustrating an exemplary derivation of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings anddynamically switching between a 5-winding-configuration motor fortrading off angular-acceleration in favor of increased angular-velocitybetween each successive winding-configuration, with N=3;

FIG. 12 is a table diagram illustrating an exemplary profile of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings in a5-winding-configuration motor for trading off angular-acceleration infavor of increased angular-velocity between each successivewinding-configuration;

FIG. 13 is a table diagram illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 5winding-configurations;

FIG. 14 is a table diagram illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 2winding-configurations, with N=3;

FIG. 15 is a table diagram illustrating an exemplary derivation of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings anddynamically switching between a 4-winding-configuration motor fortrading off angular-acceleration in favor of increased angular-velocitybetween each successive winding-configuration, with N=3;

FIG. 16 is a table diagram illustrating an exemplary profile of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings in a4-winding-configuration motor for trading off angular-acceleration infavor of increased angular-velocity between each successivewinding-configuration;

FIG. 17 is an additional table diagram illustrating an exemplaryoperation for optimizing the dynamic reconfiguration-switching using avoltage constant as a function of the winding-configuration number “WC”using 4 winding-configurations;

FIG. 18 is a table diagram summarizing FIGS. 7-17 in terms of the totaltime to ramp up to an angular velocity of 3V versus the total number ofavailable winding-configurations, where T=V/A;

FIG. 19 is a graph diagram summarizing FIGS. 7-17 in terms of the totaltime to ramp up to an angular velocity of 3V versus the total number ofavailable winding-configurations, where T=V/A;

FIG. 20 is a table diagram illustrating an exemplary derivation of a3-winding-configuration optimal switching algorithm for a final speed ofNV, where V is the maximum angular-velocity for the full voltage andtorque constant K, and N is an arbitrary multiplicative factor;

FIG. 21 is a table diagram illustrating an exemplary derivation of a(m+2)-winding-configuration optimal switching algorithm for a finalvelocity of NV, where V is the maximum angular-velocity for the fullvoltage and torque constant K, and N is an arbitrary multiplicativefactor;

FIG. 22 is a matrix diagram illustrating an exemplary tridiagonalcoefficient matrix [A];

FIG. 23 is a table diagram illustrating an exemplary profile of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings in a3-winding-configuration motor for trading off angular-acceleration infavor of increased angular-velocity between each successivewinding-configuration where X=(N+1)/2;

FIG. 24A is a block diagram illustrating a Y-connection and a DeltaConnection a brushless DC motor and/or electric motor with 3 phases;

FIG. 24B-C are block diagrams of views through rotors of an electricmotor;

FIG. 25 is a flowchart illustrating an exemplary method of an exemplaryoptimal switching algorithm;

FIG. 26 is a block diagram illustrating a DVD optical disk;

FIG. 27 is a block diagram of a servo system receiving information onupdated values of N and m via wireless communication, such as cell phonetelepathy, or Bluetooth, or GPS-location; and

FIG. 28 is a block diagram illustrating an exemplary process formonitoring the angular-velocity of an electric motor; and

FIG. 29 is a flowchart illustrating an exemplary method of optimizing adynamic reconfiguration-switching of motor windings in an electricmotor.

DETAILED DESCRIPTION OF THE DRAWINGS

The illustrated embodiments below provide mechanisms for increasingmaximum angular-velocity in an electric motor by use of a motor controlswitching circuit. The motor control switching circuit reduces the totalBack-EMF (BEMF) produced by the motor by bypassing a portion of themotor coils when high angular-velocity is needed. Although bypassing aportion of the motor coils reduces the angular-acceleration capabilityof the motor because the torque constant of the motor is reduced in theeffort to reduce the voltage constant of the motor, the motor controlswitching circuit is able to produce the necessary angular-accelerationwhen needed by switching in the previously bypassed motor coils.

FIG. 1 is a diagram illustrating the tape path of an exemplary tapetransport system 100. The tape transport system 100 illustrated in FIG.1 accepts a tape cartridge 102 containing first tape reel 104 on whichis wound a length of tape 106. The tape transport system 100 includes asecond tape reel 108, at least one tape head 110 and guide rollers 112.Tape head 110 may have Anisotropic Magneto-Resistive (AMR), GiantMagneto-Resistive (GMR), or Tunnel Magneto-Resistive (TMR) read elementsto read data and manufacturer written servo information from tape 106,and Thin Film (TF) write elements for writing data to tape 106. When thecartridge 102 is inserted into the tape transport system 100, the tape106 is automatically threaded around the rollers 112, across the tapehead 110 and onto the second tape reel 108.

Motors (not shown) operatively coupled to the reels 104 and 108 pull thetape 106 across the tape head 110 which reads/writes information to/fromthe tape in a known manner. The motors may also move the tape 106 fromone reel to another at high speed in fast forward and rewind operations.The motors may be directly coupled to first tape reel, 104 and secondtape reel, 108 or there may be a mechanical drive system between thereels and the motor(s). Whether directly coupled or coupled through amechanical drive system, the type of coupling determines a mechanicalrelationship between the motor(s) and the tape reels. The mechanicaldrive system could be for example, gears, belts, pulleys, clutches, etc.

All tape operations may occur with the tape 106 moving in eitherdirection, due to the serpentine format of the tape 106. Thus, eitherfirst tape reel 104 or 108 may serve as the supply reel or the take-upreel, depending upon the direction of the tape 106. In FIG. 1, the firsttape reel 104 within the cartridge 102 is shown serving as the tapesupply reel while the second tape reel 108 is shown serving as thetake-up reel. In the following description, the term “supply reel”refers to the reel operating as the supply reel at the present time andthe term “take up reel” refers to the reel operating as the take-up reelat the present time. In an alternate embodiment, the supply reel refersto the reel inside of the removable tape-cartridge. Moreover, the terms“supply motor” and “take-up motor” refer to the motors operativelycoupled to the supply and take-up reels, respectively. The type of tapetransport system 100 shown in FIG. 1 is for illustrative purposes onlyand the invention may be employed with other types of transport systems.

Typically, tape 106 moves at a constant linear velocity VTAPE acrosshead 110. Hence, as the radius of the outer wrap of tape R104 and R108,of reels 104 and 108, changes, the angular-velocity W104 and W108 ofreels 104 and 108 also change, per equation (1). Also, as VTAPEincreases, such as during a high speed search, W104 and W108 increaseper the following relationship:

W104*R104=W108*R108=VTAPE  (1).

Per equation (1) above, as the radius of one reel shrinks to itsminimum, which is at beginning-of-tape (BOT) for reel 108 andend-of-tape (EOT) for reel 104, that is where the respective motor isspinning at its maximum angular-velocity and generates the maximum BEMF.This maximum BEMF is further increased as VTAPE is increased from thenormal data I/O velocity to the high-speed search velocity. BEMF is theangular-velocity W of a reel motor multiplied by the voltage constantKvoltage of the motor, which is equal to the torque constant Ktorque ofthe motor when SI (metric) units are employed. It is the enclosedinvention, which reduces these two constants by using selectiveswitching to bypass motor coils, in order to reduce the BEMF:

BEMF=Kvoltage*W  (2).

The rotational acceleration capability of the motor is reduced perequation (3) when selectively bypassing motor coils because the torqueconstant Ktorque of the motor is reduced at the same time that thevoltage constant Kvoltage is reduced. Reduction of the torque constantKtorque reduces the torque provided by the motor, and that torquedivided by the rotational inertia of the motor and tape reel gives therotational acceleration of the motor and tape reel, equation (4).However, these bypassed coils may be selectively re-engaged when thathigher acceleration (or deceleration) is desired, preferably when theangular-velocity of the motor is in the range that permits an increasein back-EMF (BEMF).

Torque=Ktorque*Motor_Current  (3).

Rotational Acceleration=Torque/(Rotational Inertia of Motor+Reel)  (4).

FIG. 2 is a block diagram of a motor control or driver circuit 200 forbrushless DC motors coupled to the reels 104 and 108 for operation ofthe disclosed invention. A commutator 202 provides gate control for aset of power switches, such as FET switches 204, 205, 206, 207, 208 and209, which, in turn, connect/disconnect the motor windings 210, 212 and214 to/from a motor power supply 216 using switch 251. Sense resistor220, current sense 221, rectifier 222 and filter 223 provide currentsense signal 228 to current error amp and compensator 226.

Current error amp and compensator 226 compares current sense signal 228to current reference 227 and provides an error signal 229 to Pulse WidthModulation (PWM) modulator 224. Current error amp and compensator 226also provides servo loop compensation to ensure a stable feedback loopfor PWM modulator 224. Commutator 202 accepts hall sensor inputs HA, HB,HC from hall sensors 203A, 203B, and 203C, respectively. Commutator 202also accepts enable loop 230, Enable high velocity 231 which providesVelocity select input, PWM input 232 to control the reel motors 306 and308 (FIG. 3) using FET switches 204, 205, 206, 207, 208 and 209.Velocity switch output 235 controls velocity switches 410, 411, and 412(FIGS. 4, 5). PWM oscillator 225 also provides input to PWM 224.

FIG. 3 is an exemplary block diagram of a portion of the tape system 300in which the velocity switch system of the present invention may beincorporated. Motor driver circuits 200A and 200B are coupled to the tworeel motors 306 and 308, respectively. Reel motors 306 and 308, drivefirst tape reel 104 and second tape reel 108 respectively (FIG. 1). Hallsensors 304A and 304B are coupled to the two reel motors 306 and 308,respectively.

The output from hall sensors 304A and 304B are coupled to hall sensordetection logic 310. During normal servo operation hall sensor detectionlogic 310 decodes the output signals from hall sensors 304A and 304B toprovide motor rotation information for servo software 350. Hall sensordetection logic 310 may be implemented for example by software,firmware, hardware circuits (such as a field programmable gate array(FPGA) 314 as shown), a CPU, ASIC, etc., or a combination thereof. Servosoftware 350 processes the output from hall sensor detection logic 310using control system laws to produce primary motor control signals thatare transferred through motor assist ASIC (Application SpecificIntegrated Circuit) 355 and delivered to motor driver circuits 200A and200B. Motor assist ASIC 355 provides current control logic.

A description of the operation of servo control system for the two reelmotors 306 and 308 is given in application Ser. No. 10/223,967 entitled“Direction detection and count method for three channel commutationsensor”, filed on Aug. 8, 2002, by the assignee of the presentinvention, of which is hereby incorporated by reference in its entirety.

Servo software 350 operates within the microcode section 325 of CPU 316.Other software components, including host interface 330 and errorrecovery 335 also operate within the microcode section 325 of CPU 316.Host interface 330 provides communication between external hosts and CPU316. Error recovery 335 provides software procedures to enable CPU 316to direct operations to recover from errors that may occur duringoperation of the tape drive. In addition, a wireless communicationdevice 375, such as cell phone telepathy, Bluetooth or GPS-location, maybe used to input changes of N and m (as described below with particularreference to FIG. 25) into a servo system employing method 2500.

FIG. 4 shows a first embodiment of winding switches 410-412 with motorcoils 420-425. The winding switches 410-412 switch segments of thewindings in and out of use. The winding switches 410-412 may also bereferred to as velocity control switches. Switches 410, 411, and 412 areshown in a position to enable serial connection of motor coils 420-425.During acceleration or deceleration, velocity switch output 235activates and controls velocity switches 410, 411, and 412 in a positionto enable serial connection of motor coils 420-425. This provides themaximum torque from reel motors 306 and 308.

During periods of higher velocity, velocity switch output 235 controlsvelocity switches 410, 411, and 412 in a position to enable parallelconnection of motor coils 420-425. This provides the minimum BEMF toallow the maximum velocity from reel motors 306 and 308.

FIG. 5 shows a second embodiment of winding switches 510-512 with motorcoils 520-525. The winding switches 510-512 switch segments of thewindings in and out of use. The winding switches 510-512 may also bereferred to as velocity control switches. Switches 510, 511, and 512 areshown in a position to enable serial connection of motor coils 520-525.During acceleration or deceleration, velocity switch output 235 controlsvelocity switches 510, 511, and 512 in a position to enable serialconnection of motor coils 520-525. This provides the maximum torque fromreel motors 306 and 308.

During periods of higher velocity, velocity switch output 235 controlsvelocity switches 510, 511, and 512 in a position to enable bypass ofmotor coils 520, 522, and 524 (coils 520, 522, and 524 are left open).This provides the minimum BEMF to allow the maximum velocity from reelmotors 306 and 308.

Turning to FIG. 6, an exemplary method of operation incorporating themechanisms of the present invention is depicted. As one skilled in theart will appreciate, various steps in the method may be implemented indiffering ways to suit a particular application. In addition, thedescribed method may be implemented by various means, such as hardware,software, firmware, or a combination thereof. For example, the methodmay be implemented, partially or wholly, as a computer program productincluding a computer-readable storage medium having computer-readableprogram code portions stored therein. The computer-readable storagemedium may include disk drives, flash memory, digital versatile disks(DVDs), Blu-Ray disks, compact disks (CDs), and other types of storagemediums.

FIG. 6 shows an exemplary flowchart 600 for operation. At step 605,control circuit 200 receives a command from a tape drive to change therotation of reel motors 306 and 308. If at step 608, the tape driverequires an accelerate mode of operation, then step 612 is executed toenable velocity control switches 510-512 for serial coil connection. Ifat step 608, the tape drive requires an accelerate mode of operation,then step 611 is executed to disable velocity control switches 510-512for serial coil connection.

Control flows from step 611 or 612 to step 615. At step 615, reel motors306 and 308 are stopped, then control flows to step 630 to end,otherwise control flows to step 610, to receive another command from thetape drive.

In certain embodiments, more than two motor coils per phase may be usedto provide multiple maximum velocities for a given motor and powersupplies. For conceptual purposes, the mechanisms of the presentinvention may be thought to be analogous to a transmission in a car. Forslower speeds and more torque (to provide greater acceleration) multiplemotor coils are electronically switched in like low gears of atransmission, such as all coils 520-525 being electrically engaged asshown in FIG. 5. For higher velocities, fewer coils are used to reducethe BEMF, similar to the higher gears in a transmission, such asselectively bypassing coils 520, 522, and/or 524 as shown in FIG. 5.

The mechanisms of the present invention may be adapted for a variety oftape transport systems including a variety of tape media and tapedrives, as one skilled in the art will anticipate. While one or moreembodiments of the present invention have been illustrated in detail,the skilled artisan will appreciate that modifications and adaptationsto those embodiments may be made without departing from the scope of thepresent invention as set forth in the following claims. For example,this same invention may be applied to hard disk drives and optical diskdrives, as both of which use DC motors to spin a disk and both of whichcould benefit from faster spinning disks to reduce latency times fordata I/O. Additionally, this invention may be applied to both opticaltape as well as magnetic tape.

Moreover, the illustrated embodiments provide for optimaldynamic-reconfiguration-switching of coils within an electric motor ofeither a Y or Delta connection. More specifically, optimizing thedynamic reconfiguration-switching between individual motor windingsoccurs between multiple winding-configurations for increasingangular-velocity upon detecting the electric motor is at an optimalangular-velocity for an inductance switch, thereby preventing a totalback electromotive force (BEMF) from inhibiting further increase inangular-velocity. The dynamic reconfiguration-switching of motorwindings occurs between each of the winding-configurations at a minimal,optimal time for allowing a dynamic trade-off between theangular-velocity and angular-acceleration. In other words, the dynamicreconfiguration-switching of motor windings is optimized betweenwinding-configurations for trading off angular-acceleration in favor ofhigher angular-velocity upon detecting an electric motor is at anoptimal angular-velocity for switching to an optimal lower torqueconstant and voltage constant (the two being equal in SI units), therebypreventing a total back electromotive force (BEMF) from inhibitingfurther acceleration to a higher angular-velocity.

As illustrated below, as used in the SI (System International, commonlyknown as “metric”) units, the Voltage Constant (Kv) equals the TorqueConstant (Kt). The symbol “K” is used to denote either the VoltageConstant (Kv) or Torque Constant (Kt) when all of the motor windings arein use, since Kv and Kt are both equal when measured in SI (SystemInternational or “metric” units). For example, the equation K=Kv=Ktillustrates K is equal to the voltage constant “Kv” and the also equalto the torque constant “Kt”. “K” is used for simplicity. Also, as willbe described below, a list of Parameters used are described as:

K=voltage constant Kv=torque constant Kt (Kv=Kt for SI units),V=Maximum Angular-velocity of Motor under full voltage constant K, inradians per second,A=Angular-acceleration of Motor under full voltage constant K, inradians per secondT=V/A=units of time in seconds, for V and A defined abovem=number of equations=number of unknowns in the analysis,j=index, 1≦j≦m,X_(j)=X(j)=unknown, 1≦j≦m,N=Multiplicative factor that V is multiplied by, N>1,[A]=symmetric, tridiagonal coefficient matrix, size m-by-m,{X}=vector of unknowns X_(j), length of m,{b}=right hand side vector={1 0 0 . . . 0 N}^(T),F_(j)=F(i)=factor used in the solution of [A]{X}={b},G_(j)=G(i)=factor used in the solution of [A]{X}={b},{KF}=vector of fractional voltage and torque constants={K, K/X_(j),K/N}^(T),{Max_V}=vector of angular velocities for switching={V, X_(j)V, NV}^(T),WC=Number of possible winding-configurations in the motor=m+2, and“m” denotes the number of simultaneous linear equations and number ofunknowns. It should be noted that throughout the specification the term“winding-configuration” may also be referred to as state, switch-state,configuration-state, switch-configuration-state, andwinding-configuration-state. In all tables of the description, theinitial voltage and torque constant will always be the maximum K becauseall of the voltage and torque constant are to be used. Similarly, thefinal value of voltage and torque constant is always K divided N (e.g.,K/N). Thus, there are m+2 winding-configurations in a motor analyzed bym simultaneous linear equation and m unknowns, and the other twowinding-configurations (initial and final) are known; however, it is notknown at what angular-velocity to optimally transition between the m+2winding-configurations and vector {Max_V}, and the present inventionsolves that optimal-control challenge.

Turning now to FIG. 7, a table diagram 700 illustrating an exemplaryderivation of the optimal switching calculation is depicted foroptimizing a dynamic reconfiguration-switching between individual motorwindings and dynamically switching between a 3-winding-configurationmotor for trading off angular-acceleration in favor of increasedangular-velocity between each successive winding-configuration, withN=3. The table gives the construction of a switching algorithm, assumingthat the final angular-velocity is 3V, where V represents anundetermined number of radians per second of angular rotation and N=3 isused to give numerical answers to the calculations. The voltage constantand torque constant show that K, K/X, and K/3 represent the threewinding-configurations progressing from winding-configuration-1 whereall of the motor windings are in full use, to winding-configuration-2,and finally to winding-configuration-3 in successive order. In otherwords, the voltage constant and torque constant show a maximal value ofK, and then decrease in value to K/X, and finally to K/3 for the threewinding-configurations progressing from initial winding-configuration-1where all motor windings are used, to winding-configuration-2, andfinally to winding-configuration-3 in successive order. Theangular-acceleration starts at a maximal value of A, then decreases invalue to A/X, and finally to A/3 for the three winding-configurationsprogressing from winding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The delta angular-velocityis V, XV−V=(X−1)V, and 3V−XV=(3−X)V for the three winding-configurationsprogressing from winding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The maximalangular-velocity increases from V, XV and 3V for the threewinding-configurations progressing from winding-configuration-1,winding-configuration-2 and winding-configuration-3 in successive order.The delta time is (V/A), X*(X−1)*(V/A), and 3*(3−X)*(V/A) for the threewinding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3.

By adding up the right-most column (labeled as “Delta Time”) in FIG. 7,the total time to accelerate via the switching algorithm is expressed bythe following single, algebraic, and quadratic equation in unknown X:

$\begin{matrix}{{{Total\_ time} = {\left( \frac{V}{A} \right)*\left\lbrack {1 + {X*\left( {X - 1} \right)} + {3\left( {3 - X} \right)}} \right\rbrack}},} & {(5),}\end{matrix}$

where X is unknown coefficient and is unitless, and the algebraicexpression has the units of time from the quotient V/A, where V/A is adelta time, V is angular-velocity and A is angular-acceleration when allmotor windings are engaged, 3V is the final angular-velocity in radiansper second (used only as an example representing N=3) and X is theunknown value.

By differentiating the total time (equation 5) with respect to X, tofind the optimal value of X, and setting that derivative to zero, thefollowing linear algebraic equation in unknown X is attained:

$\begin{matrix}{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {3\left( {3 - X} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2X} - 1 - 3} \right\rbrack} = 0}},} \right.} & (6)\end{matrix}$

whereby solving for X yields the unknown value of X, and it isdetermined that X equals 2 (e.g., X=2). By taking a second derivative ofthe total time with respect to unknown X in equation 5, the secondderivative is derived to be 2V/A, which is a positive constant:

$\begin{matrix}{\frac{2V}{A} > 0.} & {(7).}\end{matrix}$

This positive-second derivative indicates the optimal time forperforming the dynamic reconfiguration-switching between individualmotor windings in order to minimize the total time to ramp up to theangular velocity 3V. Because the second derivative is positive, it meansthat the algorithm has found the value of K/X equals (a) K/2 to switchto, (b) the time and angular-velocity to switch from K to K/2, and (c)the time and angular-velocity to switch from K/2 to K/3 in order tominimize the time to ramp up to the final angular-velocity 3V radiansper second. In other words, the positive second derivative means theoptimal time to perform the dynamic switching is determined, which isindeed the optimal solution for a 3-winding-configuration motor (thewinding-configurations are defined in FIG. 8 below) going from 0 radiansper second to 3V radians per second in the minimal, hence optimal,amount of time. In terms of introductory algebra, to help visualize thisparticular solution process, the total_time in equation (5) for a threewinding-configuration motor is a simple parabola (a conic section) whichis concave, meaning that it would “hold water” like a soup bowl. Takingthe first derivative of the parabola with respect to X and setting thatfirst derivative equal to zero, plus the fact that the second derivativeof the parabola with respect to X is positive, results in the value ofX=2 where the total_time in equation (5) is minimized and thus theperformance of the three winding-configuration motor is optimized.

At this point, it is essential to introduce a generic value of “m” todenote the number of simultaneous, linear equations and to denote thenumber of unknowns, as will be used throughout the description. Also, itshould be noted that “m=1” equations and “m=1” unknowns were used toderive the table in FIG. 8, as seen below, because the initial voltageand torque constant will always be the maximum K (voltage and torqueconstant) so that no voltage and/or torque constant go unused.Similarly, the final value of the voltage and torque constant is alwaysK divided by the multiplicity “N” of angular-velocity V, hence K/3(where N=3). The calculations are only determining X to define K/X whichresulted in m=1 equation and m=1 unknown. Thus, there are m+2winding-configurations in a motor analyzed by m simultaneous linearequation and m unknowns.

FIG. 8 is a table diagram 800 illustrating an exemplary profile of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings in a3-winding-configuration motor for trading off angular-acceleration infavor of increased angular-velocity between each successivewinding-configuration. The voltage constant and torque constant show amaximal value of K, and then decrease in value to K/2, and finally toK/3 for the three winding-configurations progressing from initialwinding-configuration-1 where all motor windings are used,winding-configuration-2 and winding-configuration-3 in successive order.The angular-acceleration shows a maximal value of A, and then decreasesin value to A/2, and finally to A/3 for the three winding-configurationsprogressing from winding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The delta angular-velocityis V for each the three winding-configurations progressing fromwinding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The total angular-velocityincreases from V, to XV=2V, to 3V for the three winding-configurationsprogressing from winding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The total angular velocityis the sum of the delta angular velocities, hence the total angularvelocity for winding-configuration-2 is 2V=V+V, and the total angularvelocity for winding-configuration-3 is 3V=V+V+V. The delta time is(V/A)=T, 2(V/A)=2T, and 3(V/A)=3T giving a total “optimal” time(summation of the delta times) of 6T seconds to ramp up (e.g.,accelerate) to an angular-velocity of 3V radians per second.

FIG. 9 is a table diagram 900 illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 3winding-configurations. Corresponding to FIG. 9, FIG. 10 is a graphdiagram 1000 of FIG. 9, illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 3winding-configurations. The voltage constant and torque constant show amaximal value of K, and then decrease in value to K/2, and finally toK/3 for the three winding-configurations progressing fromwinding-configuration-1 where all motor windings are used,winding-configuration-2 and winding-configuration-3 in successive order.The angular-acceleration shows a maximal value of A, and then decreasesin value to A/2, and finally to A/3 for the three winding-configurationsprogressing from winding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The time whenwinding-configuration “WC” is in effect shows 0 to T, T to 3T, and 3T to6T for the three winding-configurations progressing fromwinding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. In other words,winding-configuration-1 is in effect from 0 to T and the motor windingsare fully engaged for a maximal torque and voltage constant of K, andthe motor is spinning up to an angular-velocity of V radians per second.When this angular-velocity V is reached, at time T, the motor windingsare dynamically switched from winding-configuration-1 towinding-configuration-2 and the motor then operates at a lower voltage(and torque) constant K/2 and lower angular-acceleration A/2 to furtherincrease its angular-velocity until it reaches a new angular-velocity 2Vat time 3T. At time 3T, the dynamic switching is made fromwinding-configuration-2 to winding-configuration-3 and the motor thenoperates at still lower voltage (and torque constant) K/3 to furtherincrease its angular-velocity until reaching a new angular-velocity 3Vis reached at time 6T. The optimal, dynamic switching occurs when thevelocity sensor of the electric motor detects that the electric motor isat the appropriate angular-velocity for such a switch in Kt and Kv.Without such a switch to reduce the voltage constant of the motor, theback-EMF of the motor would prohibit further increase of velocity beyondangular velocity V. Hence, the voltage constant of the motor is reducedto allow increased angular-velocity. Because the voltage constant andtorque constant are equal (in SI units), changes to the motortorque-constant, which is equal to its voltage-constant, have an inverserelationship with the maximal angular-velocity of the motor, namelyreducing the voltage constant increases the maximum attainable angularvelocity (inversely proportional). However, if more angular accelerationis desired, additional windings are added, as there is a 1:1relationship between the motor torque constant and angular-acceleration(directly proportional). Thus, the optimal switching algorithm allowsthe dynamic trade-off between higher angular-velocity and higherangular-acceleration.

FIG. 9 and FIG. 10 are neither geometric, powers of two, nor Fibonacciprogressions. The optimal switching algorithm could be applied to tapedrives, to make use of lower voltage power supplies, while stillachieving high speed rewind. The optimal switching algorithm could beused in hard drive (HDD) motors, to achieve higher disk angular velocitywhile reducing motor heating. These same results could be used inelectric cars, where the switching algorithm is effectively a 3-speedtransmission, by way of example only, and thus eliminating the need fora separate transmission. With the switching algorithm described above ittakes 6V/A (6T) seconds to accelerate to an angular-velocity of 3Vradians per second, as seen in FIG. 9 and FIG. 10. Without the dynamicalswitching, it would take 9V/A (9T) seconds to accelerate to the sameangular-velocity because a motor with a 3 times lower voltage constantwould have to be employed to reach an angular-velocity of 3V radians persecond. Thus, the mechanisms described herein, gain a significantadvantage by employing dynamic-reconfiguration coil-switching within aelectric motor, by using higher angular-acceleration at lowerangular-velocity, and dynamically reducing the voltage-constant to keepaccelerating, albeit at a lower angular-acceleration, to yield higherangular-velocity.

Having produced a methodology of m simultaneous equations and munknowns, as described above, FIG. 11, below, analyzes afive-winding-configuration motor, one where five different voltageconstants are used, which is a more complicated motor from thethree-winding-configurations motor analyzed above. Turning now to FIG.11, a table diagram 1100 illustrating exemplary derivation of theoptimizing a dynamic reconfiguration-switching operations betweenindividual motor windings and dynamically switching between a5-winding-configuration motor for trading off angular-acceleration infavor of increased angular-velocity between each successivewinding-configuration, with N=3, is depicted. FIG. 11 is a derivation ofm+2=5, where m=3, winding-configuration switching algorithm, and N=3 isused to give numerical answers to the calculations. The Fig.'s describedherein give the construction of a switching algorithm, assuming that thefinal angular-velocity is 3V radians per second (N=3). The voltageconstant and torque constant show a maximal value of K, followed by K/X,K/Y, K/Z, and finally to K/3 for the five winding-configurationsprogressing from winding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. The angular-accelerationshows a maximal value of A, followed by A/X, A/Y, A/Z, and thendecreases in value to A/3 for the five winding-configurationsprogressing from winding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. The delta angular-velocityis V, XV−V=(X−1)V, YV−XV=(Y−X)V, ZV−YV=(Z−Y)V, and 3V−ZV=(3−Z)V for thefive winding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3,winding-configuration-4, and winding-configuration-5 in successiveorder. The maximal angular-velocity increases from V, XV, YV, ZV, andfinally to 3V for the five winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. The delta time is (V/A),X*(X−1)*(V/A), Y*(Y−X)*(V/A), Z*(Z−Y)*(V/A), and 3*(3−Z)*(V/A) for thefive winding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3,winding-configuration-4, and winding-configuration-5 in successiveorder. By adding up the right-most column (labeled as “Delta Time”), thetotal time to accelerate via the switching algorithm is expressed by thefollowing single, algebraic, quadratic equation in unknowns X, Y, and Z:

$\begin{matrix}{\left. {{Total\_ Time} = {{\left( \frac{V}{A} \right)*\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {Y - X} \right)}} \right\rbrack} + {Z*\left( {Z - Y} \right)} + {3\left( {3 - Z} \right)}}} \right\rbrack,} & (8)\end{matrix}$

where X, Y, and Z are unknown coefficients of the total time (equation8) and are unitless, and the algebraic expression has the units of timefrom the quotient V/A, where V is the maximal angular-velocity and A isthe angular-acceleration when all motor windings are engaged, and 3V isthe final angular-velocity (N=3 used only as an example so that we getnumerical answers). The unknown coefficients X, Y, and Z are unit-lessand this algebraic expression has the units of time from the quotientV/A. (It should be noted that the variables X, Y, and Z may also beillustrated with other variable such as X₁, X₂, and X₃ asused andapplied throughout the specification.) By successively differentiatingthe total time with respect to X, Y, and then Z, to find their optimalvalues by setting the respective derivatives to zero, these m=3simultaneous linear equations (e.g., 3 simultaneous linear equations)are obtained and m=3 unknowns (e.g., 3 unknowns) are also obtained. Inother words, by differentiating the total time with respect to X, Y, andZ to find optimal values of X, Y, and Z and setting each derivative forX, Y, and Z equal to zero, these m=3 simultaneous linear equations areattained:

$\begin{matrix}{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {Y - X} \right)} + {Z*\left( {Z - Y} \right)} + {3\left( {3 - Z} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2X} - 1 - Y} \right\rbrack} = 0}},} \right.} & {(9),} \\{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {Y - X} \right)} + {Z*\left( {Z - Y} \right)} + {3\left( {3 - Z} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2Y} - X - Z} \right\rbrack} = 0}},} \right.} & {(10),} \\{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {Y - X} \right)} + {Z*\left( {Z - Y} \right)} + {3\left( {3 - Z} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2Z} - Y - 3} \right\rbrack} = 0}},} \right.} & {(11),}\end{matrix}$

whereby solving for X, Y, and Z yields an optimal value for each of X,Y, and Z. The m=3 simultaneous linear equations and m=3 unknowns tosolve are shown below, and it should be noted that these simultaneousequations take on the form of a tridiagonal matrix, which is derived ina “general” form in FIG. 21. For simplicity, using the variables X₁, X₂,and X₃ respectively in place of the variables X, Y, and Z and alsosubstituting equation 8 (total time) into the formula, these sameequations (9), (10), and (11), may appear as:

d[(V/A)*[total time]/dX ₁=(V/A)*[2X ₁ −X ₂]=0,  (9B),

d[(V/A)*[total time]/dX ₂=(V/A)*[2X ₂ −X ₁ −X ₃]=0,  (10B),

d[(V/A)*[total time]/dX ₃=(V/A)*[2X ₃ −X ₂−3]=0,  (11B),

The reason for the tridiagonal matrix is because each “interior”winding-configuration “j” in the electric motor is only affected by itsneighboring winding-configuration “j−1” and “j+1,” hence the coefficientmatrix [A] has zero values everywhere except in the main diagonal, whichis all 2's (e.g, the number “2”), and diagonals immediately adjacent tothe main diagonal, which are both all −1's (e.g., the negative number“1”). In other words, the solution to the problem that is being solveduses matrix algebra with a characteristic form, namely a tridiagonalmatrix. The coefficient matrix [A] is symmetric as well as tridiagonal.There are problems (but not in the present invention) that use“penta-diagonal” matrices and non-symmetric matrices, so being able todefine the coefficient matrix of the present invention, as bothtridiagonal and symmetric, are important mathematical properties of thepresent invention. Thus, the m=3 simultaneous linear equations and m=3unknowns to solve results in these three simultaneous equations:

2X−Y=1,  (12),

−X+2Y−Z=0,  (13),

−Y+2Z=3,  (14),

and these equations yield the solution vector as X=3/2, Y=2, and Z=5/2.This solution vector is then used in FIG. 12, below.

FIG. 12 is a table diagram 1200 illustrating an exemplary profile of theoptimal switching calculation for optimizing dynamic windingreconfiguration-switching between individual motor windings in a5-winding-configuration motor (e.g., WC=m+2=5 where m=3 as indicatedabove) for trading off acceleration in favor of increasedangular-velocity between each successive winding-configuration. Thevoltage constant and torque constants dynamically decrease from theirmaximum value of K when all motor windings are engaged, to 2K/3, to K/2,to 2K/5, and finally to K/3 for the five winding-configurationsprogressing dynamically from winding-configuration-1, towinding-configuration-2, to winding-configuration-3, towinding-configuration-4, and finally to winding-configuration-5 insuccessive order. The angular-acceleration dynamically decreases fromits maximum value of A, to 2A/3, to A/2, to 2A/5, and finally to A/3 forthe five winding-configurations progressing dynamically fromwinding-configuration-1, to winding-configuration-2, towinding-configuration-3, to winding-configuration-4, and finally towinding-configuration-5 in successive order. The delta angular-velocityis V, V/2, V/2, V/2, and V/2 for the five winding-configurationsprogressing dynamically from winding-configuration-1, towinding-configuration-2, to winding-configuration-3, towinding-configuration-4, and finally to winding-configuration-5 insuccessive order. The total angular-velocity dynamically increases fromV, to 3V/2, to 2V, to 5V/2, and finally to 3V for the fivewinding-configurations progressing dynamically fromwinding-configuration-1, to winding-configuration-2, towinding-configuration-3, to winding-configuration-4, and finally towinding-configuration-5 in successive order, and it is this increase inangular-velocity which is the desired result of the present invention.The total angular velocity is the sum of the respective delta angularvelocities, hence the total angular velocity for winding-configuration-2is 3V/2=V+V/2, the total angular velocity for winding-configuration-3 is2V=V+V/2+V/2, the total angular velocity for winding-configuration-4 is5V/2=V+V/2+V/2+V/2, and the total angular velocity forwinding-configuration-5 is 3V=V+V/2+V/2+V/2+V/2.

Taking the second derivatives, of the total time in equation (8) withrespect to X, Y, and Z (or using the variable X_(j), where j is equal to1, 2, 3 “j=1,2,3” as used and applied in the specification), allsecond-derivatives are equal to 2V/A which is positive, indicating thatthe present invention has solved for the optimal fractional voltage andtorque constants, optimal angular velocities, and optimal times toswitch to these optimal fractional voltage and torque constants.

d ²[(V/A)*[1+X*(X−1)+Y*(Y−X)+Z*(Z−Y)+3(3−Z)]/dX ²=2(V/A)>0  (15),

d ²[(V/A)*[1+X*(X−1)+Y*(Y−X)+Z*(Z−Y)+3(3−Z)]/dY ²=2(V/A)>0  (16),

d ²[(V/A)*[1+X*(X−1)+Y*(Y−X)+Z*(Z−Y)+3(3−Z)]/dZ ²=2(V/A)>0  (17),

or the following equation may be generically used where the variableX_(j) replaces the variables X, Y, and Z:

d ²[(V/A)*[total_time]/dX _(j) ²=2(V/A)>0  (18),

The unknown coefficients X, Y, and Z (and/or X_(j)) are unitless, andthe algebraic expression for total time in equation (8) has the units oftime from the quotient V/A. By taking second derivatives of the totaltime with respect to X, Y, and Z, (and/or X_(j)) a positive-secondderivative is achieved in each case, which indicates a minimal time andthus optimal time for performing the dynamic reconfiguration-switchingto achieve an angular velocity of 3V. The delta time is T, whereT=(V/A), 3T/4, T, 5T/4, and 3T/2 giving a total time of 11T/2 seconds toramp up (e.g., accelerate) to an angular-velocity of 3V radians persecond, where T is the total time, for the five winding-configurationsprogressing from winding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. Thus, with afive-winding-configuration motor (e.g., m+2=5 “five” different voltageconstants), the optimal total time to ramp up (e.g., accelerate) is11T/2 (5.5T) seconds, which is slightly faster than the 6T secondsobtained by the optimal three-winding-configuration motor, anddefinitely faster than 9T seconds obtained for a motor with no switching(a one-winding-configuration motor). FIG. 12 provides the optimalsolution for a five-winding-configuration motor, which is T/2 secondsfaster improvement in ramp-up time versus the optimal solution for thethree-winding-configuration motor.

FIG. 13 is a table diagram 1300 illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 5winding-configurations (e.g., 5 states). The voltage constant and torqueconstant shows a maximal value of K, and then decrease in value to 2K/3,K/2, 2K/5, and finally to K/3 for the five winding-configurationsprogressing from winding-configuration-1 where all motor windings areused, winding-configuration-2, winding-configuration-3,winding-configuration-4, and finally winding-configuration-5 insuccessive order. The angular-acceleration shows a maximal value of A,and then decrease in value to 2A/3, A/2, 2A/5, and finally to A/3 forthe five winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. The delta angular-velocityis V, V/2, V/2, V/2, and V/2 for the five winding-configurationsprogressing dynamically from winding-configuration-1, towinding-configuration-2, to winding-configuration-3, towinding-configuration-4, and finally to winding-configuration-5 insuccessive order. The angular-velocity successively increases from V,3V/2, 2V, 5V/2, and finally to 3V for the five winding-configurationsprogressing from winding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. The total angular velocityis the sum of the respective delta angular velocities, hence the totalangular velocity for winding-configuration-2 is 3V/2=V+V/2, the totalangular velocity for winding-configuration-3 is 2V=V+V/2+V/2, the totalangular velocity for winding-configuration-4 is 5V/2=V+V/2+V/2+V/2, andthe total angular velocity for winding-configuration-5 is3V=V+V/2+V/2+V/2+V/2. The time when winding-configuration “WC” is ineffect showing 0 to T, T to 7T/4, 7T/4 to 11T/4, 11T/4 to 4T, and 4T to11T/2 for the five winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, winding-configuration-4, andwinding-configuration-5 in successive order. In other words,winding-configuration-1 (e.g., switching-state) is from 0 to T and themotor windings are fully engaged for a voltage and torque constant of K,spinning up to angular-velocity V radians per second. When thisangular-velocity V is reached, then at time T, the motor windings aredynamically switched from winding-configuration-1 towinding-configuration-2 and the motor then operates at a lower voltage(and torque) constant 2K/3 and lower angular-acceleration 2A/3 tofurther increase its angular-velocity until it reaches angular-velocity3V/2 at time 7T/4. When this angular-velocity 3V/2 is reached, at time7T/4, the motor windings are dynamically switched fromwinding-configuration-2 to winding-configuration-3 and the motor thenoperates still at a lower voltage and torque constant K/2, andsubsequently still at a lower angular-acceleration, A/2, to furtherincrease its angular-velocity time until it reaches angular-velocity 2Vat 11T/4. When this angular-velocity 2V is reached, at time 11T/4, themotor windings are dynamically switched from winding-configuration-3 towinding-configuration-4 and the electric motor then operates still at alower voltage and torque constant, 2K/5, and subsequently still at alower angular-acceleration, 2A/5, to further increase itsangular-velocity time until it reaches angular-velocity 5V/2 at 4T. Whenthis angular-velocity 5V/2 is reached, at time 4T, the motor windingsare dynamically switched from winding-configuration-4 towinding-configuration-5 and the electric motor then operates still at alower voltage and torque constant, K/3, and subsequently still at alower angular-acceleration, A/3, to further increase itsangular-velocity time until it reaches angular-velocity 3V at 11T/2. Theoptimal, dynamic switching of individual motor windings occurs when thevelocity sensor of the motor detects that the motor is at theappropriate angular-velocity for such a switch in Kt and Kv. Withoutsuch a switch to reduce the voltage constant of the motor, the back-EMFof the motor would prohibit further increase of velocity. Hence, thevoltage constant of the motor is reduced to allow increasedangular-velocity. Because the voltage constant and torque constant areequal (in SI units), changes to the motor torque-constant, which isequal to its voltage-constant, have an inverse relationship with themaximal angular-velocity of the motor, namely reducing the voltageconstant increases the maximum attainable angular velocity (inverselyproportional). However, if more angular acceleration is desired,additional windings are added, as there is a 1:1 relationship betweenthe motor torque constant and angular-acceleration (directlyproportional). Thus, the optimal switching algorithm allows the dynamictrade-off between higher angular-velocity and higherangular-acceleration.

As the speed of the electric motor is ramped up (e.g., accelerated), themotor torque constant, which is equal to the voltage constant in SIunits and thus is equal to the angular-acceleration, monotonicallydecreases, as the angular-acceleration is traded-off for a highermaximum angular-velocity in that particular state. Thus, the msimultaneous-equations, with m unknowns (e.g., multiple unknownsrepresented by the variable “m”) as described in FIGS. 7-13, are used tosolve for the optimal switching voltage constants and determine at whatoptimal angular-velocity to perform the dynamic switching of theindividual motor windings and transition between the multiplewinding-configurations, going from one winding-configuration to the nextin a sequential order. The number of equations to solve increases,starting from one equation for a three-winding-configuration motor, andadding an additional simultaneous equation for each additionalwinding-configuration (each additional change in voltage constant) addedbeyond that to whichever winding-configuration that is desired. Thus, asillustrated in FIGS. 11 and 12, there are three linear equations withthree unknowns for the five-winding-configuration electric motor. Moregenerically for example, for performing the dynamicreconfiguration-switching of motor windings in WC=5winding-configurations and higher numbers of winding-configuration “WC”,a tri-diagonal set of equations would be used. Also, the onlyprogression followed by the voltage constants and torque constants inFIG. 12, is one of a monotonically decreasing progression.

As will be illustrated below, further analysis is provided fortwo-winding-configurations (2 different voltage constants) andfour-winding-configurations (4 different voltage constants) showingimproved performance. An optimal two-winding-configuration motor (e.g.m+2=2 where m=0) is analyzed in FIG. 14, giving a total time to reach anangular-velocity of 3V radians per second as 7T seconds, which is animprovement over the 9T seconds required by a single voltage constant(no change in voltage constant) motor. There are no equations and nounknowns here in the two-state, as there are too fewwinding-configurations (e.g., m=0 so there are no equations and nounknowns to be solved). The initial voltage and torque constant is K andthe final voltage and torque constant is K/3.

FIG. 14 is a table diagram 1400 illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 2winding-configurations, with N=3. The voltage constant and torqueconstant show a maximal value of K, and then decrease in value to K/3for the two winding-configurations progressing fromwinding-configuration-1 to winding-configuration-2 in successive order.The angular-acceleration shows a maximal value of A, and then decreasesin value to A/3 for the two winding-configurations progressing fromwinding-configuration-1 to winding-configuration-2 in successive order.The total angular-velocity increases from V to 3V for the twowinding-configurations progressing from winding-configuration-1 towinding-configuration-2 in successive order. The time whenwinding-configuration “WC” is in effect shows the dynamic switching ofthe motor windings occurring between winding-configuration-1, which is 0to T, and winding-configuration-2, which is T to 7T, for the twowinding-configurations progressing from winding-configuration-1 towinding-configuration-2 in successive order. In other words,winding-configuration-1 is from 0 to T and the motor windings are fullyengaged, spinning up to angular-velocity V radians per second. When thisangular-velocity V is reached, then at time T, the motor windings aredynamically switched from winding-configuration-1 towinding-configuration-2 and the electric motor then operates at a lowervoltage (and torque) constant K/3 and lower angular-acceleration A/3 tofurther increase its angular-velocity until it reaches angular-velocity3V at time 7T. The optimal, dynamic switching occurs when the velocitysensor of the electric motor detects that the electric motor is at theappropriate angular-velocity for such a switch in Kt and Kv. Withoutsuch a switch to reduce the voltage constant of the motor, the back-EMFof the motor would prohibit further increase of velocity. Hence, thevoltage constant of the motor is reduced to allow increasedangular-velocity. Because the voltage constant and torque constant areequal in SI units, changes to the motor torque-constant (which is equalto its voltage-constant) are directly proportional with theangular-acceleration of the motor and inversely proportional to thetotal angular velocity. Thus, the optimal switching algorithm allows thedynamic trade-off between higher angular-velocity and higherangular-acceleration to reach the maximum angular velocity in theminimal amount of time.

Turning now to FIG. 15, a table diagram 1500 illustrating an exemplaryderivation of the optimal switching calculation is depicted foroptimizing a dynamic reconfiguration-switching between individual motorwindings and dynamically switching between a 4-winding-configurationmotor for trading off angular-acceleration in favor of increasedangular-velocity between each successive winding-configuration, withN=3. FIG. 15 is a derivation of m+2=4 winding-configurations opticalswitching algorithm, where m=2. FIG. 15 analyzes afour-winding-configuration motor, where four different voltage constantsare used, which is a more complicated motor from thethree-winding-configuration motor analyzed above. The table gives theconstruction of a switching algorithm, assuming that the finalangular-velocity is 3V radians per second. The voltage constant andtorque constant show a maximal value of K, and then decrease in value toK/X, K/Y, and finally to K/3 for the four winding-configurationsprogressing from winding-configuration-1, winding-configuration-2,winding-configuration-3, and winding-configuration-4 in successiveorder. The angular-acceleration shows a maximal value of A, and thendecreases in value to A/X, A/Y, and finally to A/3 for the fourwinding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The delta angular-velocityis V, XV−V=(X−1)V, YV−XV=(Y−X)V, 3V-YV=(3−Y)V for the fourwinding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The maximalangular-velocity increases from V, XV, YV, and finally to 3V for thefour winding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The delta time is (V/A),X*(X−1)*(V/A), Y*(Y−X)*(V/A), and 3*(3−Z)*(V/A) for the fourwinding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. (It should be noted thatthe variables X and Y may also be illustrated with other variable suchas X₁, and X₂.X₁ and X₂)

By adding up the right-most column (labeled as “Delta Time”), the totaltime to accelerate via the switching algorithm is expressed by thefollowing simple, algebraic, and quadratic equation in X and Y:

$\begin{matrix}{\left. {{Total\_ Time} = {{\left( \frac{V}{A} \right)*\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {X - 1} \right)}} \right\rbrack} + {Y*\left( {Y - X} \right)} + {3\left( {3 - Y} \right)}}} \right\rbrack,} & (19)\end{matrix}$

where X and Y are unknown coefficient of the total time and areunitless, and the algebraic expression for total time has the units oftime from the quotient V/A, V is the maximal angular-velocity and A isthe angular-acceleration when all motor windings are being used, 3Vradians per second is the final angular-velocity (N=3 is used only as anexample). By successively differentiating the total time with respect toX and Y to find optimal values of X and Y and setting the derivativesfor X and Y equal to zero, these m=2 equations (e.g., 2 equations) andm=2 unknowns are attained:

$\begin{matrix}{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {Y - X} \right)} + {3\left( {3 - Y} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2X} - 1 - Y} \right\rbrack} = 0}},} \right.} & {(20),} \\{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {Y\left( {Y - X} \right)} + {3\left( {3 - Y} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2Y} - X - 3} \right\rbrack} = 0}},} \right.} & {(21),}\end{matrix}$

whereby solving for X and Y yields an optimal value for X and Y. Theunknown coefficients X and Y are unitless and these algebraicexpressions have the units of time from the quotient V/A. By takingsecond derivatives of the total time with respect to X and Y, apositive-second derivative (2V/A) is achieved in each case, whichindicates a minimal-optimal time for performing the dynamicreconfiguration-switching. For simplicity, using the variables X₁, andX₂, X₁ and X₂ respectively in place of the variables X and Y and alsosubstituting equation 19 (total time) into the formula, these sameequations (20) and (21), may appear as:

d[(V/A)*[total_time]/dX ₁=(V/A)*[2X ₁−1−X ₂]=0,  (20B),

d[(V/A)*[total_time]/dX ₂=(V/A)*[2X ₂ −X ₁−3]=0,  (21B),

and the m=2 simultaneous equations to solve are only the top and bottomrows of the general tridiagonal matrix solved for in the table of FIG.21, which results in the two simultaneous equations to solve as:

2X−Y=1,  (22),

−X+2Y=3,  (23),

and these equations yield a solution vector as X=5/3 and Y=7/3. Thissolution vector is then used in FIG. 16, below.

FIG. 16 is a table diagram 1600 illustrating an exemplary profile ofoptimizing a dynamic reconfiguration-switching between individual motorwindings in a 4-winding-configuration (e.g., m+2=4, where m=3, asdescribed above) motor for trading off acceleration in favor ofincreased angular-velocity between each successivewinding-configuration. The voltage constant and torque constant show amaximal value of K, and then decrease in value to 3K/5, 3K/7, andfinally to K/3 for the four winding-configurations progressing fromwinding-configuration-1 when all motor windings are used,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The angular-accelerationshows a maximal value of A, and then decreases in value to 3A/5, 3A7,and finally to A/3 for the four winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, and winding-configuration-4 in successiveorder. The delta angular-velocity is V, 2V/3, 2V/3, and 2V/3 for thefour winding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The total angular velocityincreases from V, 5V/3, 7V/3, and finally to 3V for the fourwinding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The total angular velocityis the sum of the respective delta angular velocities, hence the totalangular velocity for winding-configuration-2 is 5V/3=V+2V/3, the totalangular velocity for winding-configuration-3 is 7V/3=V+2V/3+2V/3, thetotal angular velocity for winding-configuration-4 is3V=V+2V/3+2V/3+2V/3. The delta time is T, where T=(V/A), 10T/9, 14T/9,and 2T giving a total time of 51T/9 seconds to ramp up (e.g.,accelerate) to an angular-velocity of 3V radians per second, where 51T/9is the total time for the four winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, and winding-configuration-4 in successiveorder. Taking second derivatives of the total time for X and Y, bothsecond-derivatives are positive (and equal to each other), indicatingthe solutions of X and Y provide the optimal fractional voltage andtorque constants, optimal angular velocities, and optimal times toswitch to these optimal fractional voltage and torque constants:

d ²[(V/A)*[1+X*(X−1)+Y*(Y−X)+3(3−Y)]/dX ²=2(V/A)>0  (24),

d ²[(V/A)*[1+X*(X−1)+Y*(Y−X)+3(3−Y)]/dY ²=2(V/A)>0  (25),

the following equation may be used generically where the variable X_(j)(where j=1 or 2) replaces the variables X and Y, as in equation (19) and(20):

d ²[(V/A)*[total_time]/dX _(j) ²=2(V/A)>0  (26),

thus, with a four-winding-configuration motor (four different voltageconstants), the optimal total time to ramp up (e.g., accelerate) is51T/9 (5.67T) seconds which is slower than 11T/2 (5.5T) seconds for afive-winding-configuration motor, but faster than the 6T secondsobtained by the optimal three-winding-configuration motor, and alsofaster than 9T seconds obtained for a motor with no switching (aone-winding-configuration motor). To visualize this particular solutionprocess, the total_time in equation (19) for a fourwinding-configuration motor is a parabolic surface (like the reflectivesurface of a telescope mirror or an automobile headlight) which isconcave, meaning that it would “hold water” like a soup bowl. Taking thefirst derivatives of the parabola with respect to X and Y and settingthose first derivatives equal to zero, plus the fact that the secondderivatives of the parabola with respect to X and Y are positive,results in the values of X=5/3 and Y=7/3 where the total_time inequation (19) is minimized and thus the performance of the fourwinding-configuration motor is optimized.

FIG. 17 is a table diagram 1700 illustrating an exemplary operation foroptimizing the dynamic reconfiguration-switching using a voltageconstant as a function of the winding-configuration number “WC” using 4winding-configurations (e.g., 4 switching-states). As the speed of theelectric motor is ramped up, the motor torque constant, which is equalto the voltage constant in SI units and thus is equal to theangular-acceleration, monotonically decreases, as theangular-acceleration is traded-off for a higher maximum angular-velocityin that particular state. The voltage constant and torque constant showa maximal value of K, and then decrease in value to 3K/5, 3K/7, andfinally to K/3 for the four winding-configurations progressing fromwinding-configuration-1 where all motor windings are used,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The angular-accelerationshows a maximal value of A, and then decrease in value to 3A/5, 3A/7,and finally to A/3 for the four winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, and winding-configuration-4 in successiveorder. The delta angular-velocity is V, 2V/3, 2V/3, and 2V/3 for thefour winding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The total angular-velocityincreases from V, 5V/3, 7V/3, finally to 3V for the fourwinding-configurations progressing from winding-configuration-1,winding-configuration-2, winding-configuration-3, andwinding-configuration-4 in successive order. The total angular velocityis the sum of the respective delta angular velocities, hence the totalangular velocity for winding-configuration-2 is 5V/3=V+2V/3, the totalangular velocity for winding-configuration-3 is 7V/3=V+2V/3+2V/3, thetotal angular velocity for winding-configuration-4 is3V/=V+2V/3+2V/3+2V/3. The time when winding-configuration “WC” is ineffect shows the dynamic switching of the motor windings occurringbetween winding-configuration-1, which is 0 to T, andwinding-configuration-2, which is T to 19T/9, andwinding-configuration-2 and winding-configuration-3, which is 19T/9 to33T/9, and winding-configuration-3 and winding-configuration-4, which is33T/9 to 51T/9, for the four winding-configurations progressing fromwinding-configuration-1, winding-configuration-2,winding-configuration-3, and winding-configuration-4 in successiveorder. In other words, winding-configuration-1 (e.g., switching-state)is from 0 to T and the motor windings are fully engaged, resulting in atorque constant voltage constant of K, spinning up to angular-velocity Vradians per second. When this angular-velocity V is reached, then attime T, the motor windings are dynamically switched fromwinding-configuration-1 to winding-configuration-2 and the electricmotor then operates at a lower voltage (and torque) constant 3K/5, andlower angular-acceleration 3A/5, to further increase itsangular-velocity until it reaches angular-velocity 5V/3 at time 19T/9.When this angular-velocity 5V/3 is reached, at time 19T/9, the motorwindings are dynamically switched from winding-configuration-2 towinding-configuration-3 and the electric motor then operates still at alower voltage and torque constant 3K/7, and subsequently still at alower angular-acceleration, 3A/7, to further increase itsangular-velocity time until it reaches angular-velocity 7V/3 at 33T/9.When this angular-velocity 7V/3 is reached, at time 33T/9, the motorwindings are dynamically switched from winding-configuration-3 towinding-configuration-4 and the electric motor then operates still at alower voltage and torque constant, K/3, and subsequently still at alower angular-acceleration, A/3, to further increase itsangular-velocity time until it reaches angular-velocity 3V at 51T/9. Theoptimal, dynamic switching of individual motor windings occurs when thevelocity sensor of the motor detects that the motor is at theappropriate angular-velocity for such a switch in Kt and Kv.

FIG. 18 is a table diagram summarizing FIGS. 7-17 in terms of the totaltime to ramp up to an angular velocity of 3V versus the total number ofavailable winding-configurations, where T=V/A. FIG. 19 is a graphdiagram summarizing FIGS. 7-17 in terms of the total time to ramp up toan angular velocity of 3V versus the total number of availablewinding-configurations, where T=V/A. For a motor with only onewinding-configuration, the total ramp up to 3V radians per second (themultiplication of V by N=3 is only used only as an example) is 9Tseconds, as shown in FIG. 10. In decimal form this equates to 9T. For amotor with two winding-configurations (2 switching-states), the totalramp up time (acceleration phase) to 3V radians per second is 7Tseconds, as shown in FIG. 14. In decimal form this equates to 7T. For amotor with three winding-configurations (e.g., 3 switching-states), thetotal ramp up to 3V radians per second is 6T seconds, as shown in FIG.8. In decimal form this equates to 6T. For a motor with fourwinding-configurations (e.g., 4 switching-states), the total ramp up to3V radians per second is 51T/9 seconds, as shown in FIG. 16. In decimalform this equates to 5.67T. For a motor with five winding-configurations(e.g., 5 switching-states), the total ramp up to 3V radians per secondis 11T/2 seconds, as shown in FIG. 12. In decimal form this equates to5.5T. Thus, the more winding configurations that are physicallyavailable for switching, the lower the total time to accelerate to thefinal angular velocity of 3V radians per second. However, theincremental change in total time to accelerate to the final angularvelocity of 3V radians per second diminishes as more windingconfigurations are added.

As illustrated in column 1 (labeled as the total number of WC), thetotal number of available winding configurations are 1, 2, 3, 4, and 5.As stated above, the total time for a motor only having 1 windingconfiguration is 9T. But a motor with 2 winding configurations has atotal time of 7T. Thus, as more available winding configurations areadded to the motor, the faster it reaches its 3V angular velocity. Theidea here is that as the motor has more winding configurations to choosefrom, the total time is reduced, and this is illustrated in both FIGS.18 and 19. An analogy to a car having a manual transmission may be usedfor illustration purposes. For example, if a user desires to acceleratefaster to 75 mph in the car, the car having a 4-speed transmission wouldreach the desired speed faster than a car having a 3-speed transmission.Similarly, the car having a 5-speed transmission would reach the desiredspeed faster than a car having a 4-speed transmission. This analogy isexactly what FIGS. 18 and 19 are illustrating with the motor having thevarious winding configurations.

Turning now to FIG. 20, a table diagram 2000 illustrating an exemplaryderivation of a 3-winding-configuration optimal switching algorithm foroptimizing a dynamic reconfiguration-switching between individual motorwindings and dynamically switching between a 3-winding-configurationmotor for trading off acceleration in favor of increasedangular-velocity between each successive winding-configuration isdepicted, now generalizing with N being a variable rather than thespecial case of N=3. FIG. 20 provides the construction of the switchingalgorithm, assuming that the final angular-velocity is “NV” radians persecond (N being an arbitrary valued multiplier. N previously had been 3,as illustrated above, to allow calculations to have numerical answers.)and m+2=3 (e.g., three) winding-configurations are used to illustrate ageneralization of the optimal switching algorithm for any high speedangular-velocity, rather than the 3V radians per second, as illustratedabove. For example, the use of 3V (e.g., N=3) came from the IBM® 3480tape drive which had a high speed rewind three times that of a normalread-write velocity.

By adding up the right-most column (labeled as “Delta Time”), the totaltime to accelerate via the switching algorithm is expressed by thefollowing single, algebraic, and quadratic equation in unknown X:

$\begin{matrix}{{{Total\_ time} = {\left( \frac{V}{A} \right)*\left\lbrack {1 + {X*\left( {X - 1} \right)} + {N\left( {N - X} \right)}} \right\rbrack}},} & {(27),}\end{matrix}$

where X is unknown coefficient and is unitless, and the algebraicexpression for total_time has the units of time from the quotient V/A, Vis the maximal angular-velocity, and A is the angular-acceleration whenall motor windings are being used, N is an arbitrary value greater thanunity (e.g., N=3 as used above for illustration purposes for theprevious calculations) representing a multiplier of the angular-velocityV to give the final angular-velocity as NV, and X is an unknowncoefficient.

By differentiating the total time with respect to X, to find the optimalvalue of X, and setting that derivative to zero, the following linearequation is attained:

$\begin{matrix}{\left\lbrack {{{\left( \frac{V}{A} \right)*\frac{\left\lbrack {1 + {X*\left( {X - 1} \right)} + {N\left( {N - X} \right)}} \right\rbrack}{x}} = {{\left( \frac{V}{A} \right)*\left\lbrack {{2X} - 1 - N} \right\rbrack} = 0}},} \right.} & {(28),}\end{matrix}$

whereby solving for X yields the unknown value of X, it is determinedthat X equals (N+1)/2. By taking a second derivative of the total timewith respect to X, the second derivative is derived to be:

$\begin{matrix}{{\frac{2V}{A} > 0},} & {(29),}\end{matrix}$

whereby this positive-second derivative which is achieved, indicatesthat the solution for X provides an optimal time for performing thedynamic reconfiguration-switching of the motor windings to achieve theminimal time to achieve an angular-velocity of NV. Because the secondderivative is positive, the minimum time (e.g., the minimal-optimaltime) to perform the dynamic reconfiguration-switching of the motorwindings is determined, which is indeed the optimal solution for a3-winding-configuration motor going from 0 radians per second to NVradians per second. In other words, because the second derivative ispositive, the second derivative with respect to X means (a) the value ofthe K/X equals 2K/(N+1) to switch to, (b) the time and angular-velocityto switch from K to 2K/(N+1), and (c) the time and angular-velocity toswitch from 2K/(N+1) to K/N are optimal, in order to minimize the timeto ramp up to the final angular-velocity NV is determined. These resultsare consistent with above calculations where N=3 was used. In terms ofintroductory algebra, to help visualize this particular solutionprocess, the total_time in equation (27) for a threewinding-configuration motor is a simple parabola (a conic section) whichis concave, meaning that it would “hold water” like a soup bowl. Takingthe first derivative of the parabola with respect to X and setting thatfirst derivative equal to zero, plus the fact that the second derivativeof the parabola with respect to X is positive, results in the value ofX=(N+1)/2 where the total time in equation (27) is minimized and thusthe performance of the three winding-configuration motor is optimized.This value of X=(N+1)/2 is equivalent to the value of X=2 for thespecial case of N=3 for the total_time equation (5).

FIG. 21 is a table diagram 2100 illustrating an exemplary and fullygeneralized derivation of a (m+2)-winding-configuration optimalswitching algorithm for a final velocity of NV, where V is the maximumangular-velocity for the full voltage and torque constant K. By addingup the right-most column in the table of FIG. 21, the total time toaccelerate via our switching algorithm is expressed by this singlealgebraic expression for total time which is quadratic in each unknownX₁, X₂, X₃, . . . , X_(m):

Total Time=(V/A)*[1+X ₁*(X ₁−1)+X _(2*() X ₂ −X ₁)+X ₃*(X ₃ −X ₂)+ . . .+X _(m)*(X _(m) −X _(m−1))+N*(N−X _(m))]  (30),

where {X} is the vector of unknown X₁, X₂, X₃, . . . , X_(m) that areunitless, and this single algebraic expression for total time has theunits of time from the quotient V/A, and N is an arbitrary multiplierrepresenting the final angular-velocity NV, and X is an unknowncoefficient of the total time. By successively differentiating thetotal_time with respect to X₁, X₂, X₃, . . . X_(m), to find theiroptimal values by setting the respective derivatives to zero, these msimultaneous equations and m unknowns are attained:

d[(V/A)*[Total_Time]/dX ₁=(V/A)*[2X ₁−1−X ₂]=0  (31),

d[(V/A)*[Total_Time]/dX _(j)=(V/A)*[2X _(j) −X _(j−1) −X _(j+1)]=0 forj=2 . . . m−1  (32),

d[(V/A)*[Total_Time]/dX _(m)=(V/A)*[2X _(m) −X _(m−1) −N]=0  (33),

and taking second derivatives, of total time for X₁, X₂, X₃, . . .X_(m), all second-derivatives are positive (and equal to each other),indicating that the present invention has solved for the optimalfractional voltage and torque constants, optimal angular velocities, andoptimal times to switch to these optimal fractional voltage and torqueconstants, by using the following equations for the taking the secondderivative, which are positive (e.g., greater than zero):

d ²[(V/A)*[Total_Time]/dX ₁ ²=2(V/A)>0  (34),

d ²[(V/A)*[Total_Time]/dX _(j) ²=2(V/A)>0 for j=2 . . . m−1  (35),

d ²[(V/A)*[Total_Time]/dX _(m) ²=2(V/A)>0  (36),

or the following equation may generically used where the variable X_(j)is used and replaces the variables X₁, X_(j), X_(m), as used inequations (34), (35), and (36):

d ²[(V/A)*[Total_Time]/dX _(j) ²=2(V/A)>0 for j=1 . . . . m  (37).

The m simultaneous linear equations to solve for the m unknowns are(31), (32), and (33). These m simultaneous linear equations form atridiagonal system of equations where all entries of the coefficientmatrix are 2 along the main diagonal, and −1 along the diagonalsimmediately adjacent the main diagonal, and all other entries in thecoefficient matrix are zero. As stated before, coefficient matrix [A] issymmetric as well as tridiagonal, meaning that coefficient matrix [A]has specific mathematical characteristics. The m simultaneous linearequations to solve for the m unknowns are:

2X ₁ −X ₂=1,  (38),

−X _(j−1)+2X _(j) −X _(j+1)=0 for j=2 . . . m−1,  (39),

−X _(m−1)+2X _(m) =N,  (40),

where these simultaneous linear equations to be solved are placed in thetridiagonal coefficient matrix [A], as seen in FIG. 22.

FIG. 22 is a matrix diagram 2200 illustrating an exemplary tridiagonalcoefficient matrix [A]. Here, [A]{X}={b} is a set of m linear equationsand m unknowns. The tridiagonal coefficient matrix [A] organizes the msimultaneous-linear equations having the m number of unknown variablesinto a tridiagonal system of linear equations, a coefficient matrix ofthe tridiagonal system of linear equations having 2's along an entiremain diagonal of the coefficient matrix, negative 1's along diagonalsimmediately adjacent to the main diagonal, all other entries of thecoefficient matrix being zero, and a right-hand-side vector {b}comprising a first entry of 1, a last entry of N denoting NV, which is Ntimes a maximum allowable angular-velocity V of the motor at a onehundred percent torque constant K, and all other entries of theright-hand-side vector {b} being zero. The reason for the tridiagonalmatrix is because each “interior” winding-configuration “j” in theelectric motor is only affected by its neighboring winding-configuration“j−1” and “j+1,” hence the coefficient matrix [A] has zero valueseverywhere except in the main diagonal, which is all 2's (e.g., thenumber “2”), and diagonals immediately adjacent to the main diagonal,which are both all −1's (e.g., the negative number “1”). All otherelements in coefficient matrix [A] are zero. In other words, thesolution to the problem that is being solved uses matrix algebra with acharacteristic form, namely a tridiagonal matrix. The coefficient matrix[A] is symmetric as well as tridiagonal. X₁, X₂, X₃, . . . X_(m) areshown as the unknown variable of X, and the right hand side vector {b}shows 1, 0, 0, 0, 0, N respectively. It should be noted that allexamples described herein are special cases of this generalized matrixapproach. For example, the first derivation for tables of FIGS. 11-13,N=3 and m=3 (m+2=5 winding-configuration motor). Please note that m isan independent integer and N has any positive value greater than unity(N need not be an integer).

One of the values of coefficient matrix [A] being both symmetric andtridiagonal is that a simple solution algorithm can be written, onedecidedly less complicated than if coefficient matrix [A] was fullypopulated with non-zero elements which did not fit any pattern. Forexample, a solution algorithm may begin with the calculation of factorsF_(j) and The order of the progression is starting from index j equals 1and ending at j equals m. In other words, the calculation of factorsF_(j) and G_(j) are calculated by the following recursive algorithm:

F ₁=2 and G ₁=½,  (41),

F _(j)=2−1/F _(j−1) from j=2 to m,  (42),

G _(j) =G _(j−1) /F _(j) from j=2 to m−1, and  (43),

G _(m)=(N+G _(m−1))/F _(m),  (44),

as will be shown below in FIG. 25.

The elements of solution vector {X_(j)} are now obtained, with the orderof the progression in the reverse direction, namely starting from indexj equals m and ending at j equals 1, X_(m)=G_(m), X_(j)=G, +X_(j+1)/2.Thus, the solution vector {X_(j)} is now in a simple algorithm, for m≧2(e.g., m is greater than and/or equal to two), since that is the lowestvalue of m yielding a coefficient matrix [A] with multiple simultaneouslinear equations and multiple unknowns. If m=1, there is only oneequation to be solved and the above upper-triangularization andback-substitution algorithm is unnecessary. Also, the solution for m=1from tables as seen in FIG. 18-20, is already obtained, being X=(N+1)/2.

At this point, a need exists to analyze the case for m=1 equation andm=1 unknown. The following Fig.'s give the construction of the switchingalgorithm, assuming that the final angular-velocity is NV (N is anundetermined constant greater than unity) and m+2=3winding-configurations are used, and there is only m=1 equation and m=1unknown, hence the tridiagonal matrix approach of FIG. 21 is not usedfor a simple algebra problem. FIG. 23, below, is established with thehelp of FIG. 20, which has already been discussed.

FIG. 23 is a table diagram 2300 illustrating an exemplary profile of theoptimal switching calculation for optimizing a dynamicreconfiguration-switching between individual motor windings in a3-winding-configuration motor for trading off acceleration in favor ofincreased angular-velocity between each successive winding-configurationwhere X=(N+1)/2. In FIG. 23, the variable X, which is and unknowncoefficient and is unitless, and the algebraic expression for total timehas the units of time from the quotient V/A. The voltage constant andtorque constant show a maximal value of K, and then decrease in value to2K/(N+1), and finally to K/N for the three winding-configurationsprogressing from winding-configuration-1 where all motor windings areused, winding-configuration-2 and winding-configuration-3 in successiveorder. The angular-acceleration shows a maximal value of A, and thendecreases in value to 2A/(N+1), and finally to A/N for the threewinding-configurations progressing from winding-configuration-1,winding-configuration-2 and winding-configuration-3 in successive order.The delta angular-velocity is V, (N−1)V/2, and (N−1)V/2 for the threewinding-configurations progressing from winding-configuration-1,winding-configuration-2 and winding-configuration-3 in successive order.The total angular-velocity increases from V, to (N−1)V/2, and finally NVfor the three winding-configurations progressing fromwinding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order. The total angular velocityis the sum of the respective delta angular velocities, hence the totalangular velocity for winding-configuration-2 is (N+1)V/2=V+(N−1)V/2, andthe total angular velocity for winding-configuration-3 isNV=V+(N−1)V/2+(N−1)V/2. The delta time is

${\left( \frac{V}{A} \right) = T},\frac{\left( {N^{2} - 1} \right)T}{4},{{and}\mspace{14mu} \frac{\left( {N^{2} - N} \right)T}{2}},$

giving a total time to accelerate up to an angular-velocity of NV as

$\frac{\left( {{3N^{2}} - {2N} + 3} \right)T}{4},$

which is faster than the N²T required without coil switching, again,where

${T = {\frac{V}{A}\mspace{14mu} {and}\mspace{14mu} \frac{\left( {{3N^{2}} - {2N} + 3} \right)T}{4}}},$

is the total time, for the three winding-configurations progressing fromwinding-configuration-1, winding-configuration-2 andwinding-configuration-3 in successive order.

Thus, as illustrated in Fig.'s above, in one embodiment, the presentinvention performs this process for a motor withWC-winding-configurations, where WC=m+2, of the electric motor in orderto optimally achieve an angular-velocity NV which is N times acapability of the electric motor with a one hundred percent torqueconstant and voltage constant K. Calculating an optional time forperforming the dynamic reconfiguration-switching by first finding atotal time for acceleration to an angular velocity of NV anddifferentiating that total time with respect to vector of unknowns {X}and setting each of those first derivatives to zero to find an optimalvalue of the vector of unknowns {X} to minimize the total time. Finally,taking a second derivative of that total time with respect to vector ofunknowns {X}, wherein a positive-second derivative indicates that thevalue of the vector of unknowns {X} indeed minimizes and optimizes thetotal time for performing the dynamic reconfiguration-switching, wherevector of unknowns {X} represents one or more unknowns used forsubdividing the one hundred percent torque constant and voltage constantof the electric motor into smaller units of a torque constant and avoltage constant (Kt=Kv in SI units) which allows the electric motor togo faster than V angular-velocity and up to an angular-velocity of NVwhere V is a maximum angular-velocity achieved with the one hundredpercent torque constant, and N is an arbitrary value greater than 1,where WC is the a number of possible winding-configurations of theelectric motor, and WC=m+2, where m is the number of equations andnumber of unknowns being solved for.

FIG. 24A is a block diagrams 2400 showing wye and delta connection(e.g., Y-connection and a Delta Connection) for a brushless dc motorand/or an electric motor. FIG. 24B-C are block diagrams of views throughrotors of an electric motor. As mentioned previously, the optimaldynamic-reconfiguration switching of individual motor winds (e.g.,coils) within an electric motor may be performed with either a wye (Y)or a delta connection. FIG. 24A illustrates both the wye and deltaconnection. For example, in one embodiment, by way of example only, theelectric motor may include a stator, including three windings, spaced at120 degree electrical from one another, for imparting a torque on arotor. The torque imparted on the rotor causes the rotor to rotate.Those skilled in the art of motors and generators appreciate theefficiency, economy and simplicity of electric motors wherein there isno actual physical contact between the stator windings and the rotor. Inorder to effectuate the operation of the motor, a properly timed andspaced magnetic field is synthesized in the stator windings, whichimparts a torque on the rotor and causes the rotor to rotate. Theelectric motor may be permanently configured in one of two basicconfigurations, either a wye connection or a delta connection, as seenin FIG. 24A. The electric motor with windings configured in the deltaconfiguration can operate at a greater speed than the same windingsconfigured in the wye configuration. However, the electric motor withwindings configured in the wye configuration can operate with a greatertorque at low speeds than the same windings configured in the deltaconfiguration.

Moreover, the illustrated embodiments as described in herein may beapplied to a variety of types of electric motors (e.g., an electricmotor used in the automotive industry). For example, in one type ofelectric motor, the electric motor may be comprised of an armaturebearing three windings. The armature rotates in a magnetic field andcurrent is generated in the three windings and drawn from them in turnthrough brushes. Such an electrical motor is shown in FIG. 24B. In oneembodiment, a motor armature comprises a rotor having threeequiangularly spaced poles 1 about each of which is wound a coil windingΦ. Coil windings Φ are connected to commutator segments 2 which in turnare contacted by brushes (not shown), Such a motor may be used to causerotation by applying current to the windings, which then rotate within amagnetic field, or may be used in reverse to generate current fromrotation of the windings within the magnetic field. For example, a motorwith three coil windings Φ1, Φ2 and Φ3 are all identical and haveidentical numbers of turns in each winding. When the motor rotates, thecurrent that flows through each winding is therefore identical. One wayof providing a feedback control signal is to form the three coilwindings Φ with a differing numbers of turns. This is shown in FIG. 24Bwhere winding Φ1-1 is formed with a reduced number of turns incomparison with windings Φ2 and Φ3, The effect of this is that as themotor rotates the current that flows through winding Φ1-1 is differentfrom that which flows through windings Φ2 and Φ3. This difference can bedetected and used to count the number of rotations of the motor, andalso to mark and define the beginning of rotation cycles of the motor.This information can be used in a number of ways to accurately controlthe rotation of the motor in a number of applications as discussedabove. The embodiment as described above has three poles, however itwill be understood that this is by no means essential and the motor mayhave any odd number of poles. For example, as illustrated in FIG. 24C,an electric motor may have a more than three poles (e.g., five poles).The electric motor may be comprised with multiple coil windings. Thewindings may also have an equal number of turns on one winding, Any oneor more of these poles could be provided with a reduced number of turnscompared to a regular coil winding. In short, the illustratedembodiments may be applied to a variety of types and variations ofelectric motors. It should be noted that an electric motor has beendescribed in FIG. 24B-C by way of example only and such rotors/winding,as used in the present invention, are not limited to the electric motor,but other motors commonly used in the art having the variousrotors/windings may also be applied to accomplish the spirit andpurposes of the present invention.

FIG. 25 is a flowchart illustrating an exemplary method 2500 of anexemplary optimal switching algorithm. The optimizing switchingalgorithm solutions, as seen above in FIGS. 20, 21, and 23 aresummarized in method 2500, where the algorithm is solving for {X}, {KF},and {Max_V}. The method 2500 begins (step 2502). The method 2500 inputsan initial value for N, the multiplicative factor by which V ismultiplied, where N is greater than one “1” (step 2504). The method 2500inputs an initial value for the number of equations “m,” where m isgreater than one “1” (step 2506). The method 2500 determines if m isequal to one (step 2508). If no, the method 2500 calculates thecoefficients from j equals one “1” (e.g., j=1) up to m (step 2510). Asillustrated above in FIG. 22, the calculation of factors F_(j) and G_(j)are calculated by the following recursive algorithm:

F ₁=2 and G ₁=½,  (41),

F _(j)=2−1/F _(j−1) from j=2 to m,  (42),

G _(j) =G _(j−1) /F _(j) from j=2 to m−1, and  (43),

G _(m)=(N+G _(m−1))/F _(m)  (44).

From step 2510, the method 2500 then calculates the solution vector {X}starting from j=m and going down to j=1, X(m)=G(m) andX(j)=G(j)+X(j+1)/2 (step 2512). Returning to step 2508, if yes themethod 2500 the variable X(1) is set equal to (N+1)/2 (step 2514). Fromboth steps 2512, and 2514, the method 2500 calculates the vector {KF} offractional voltage and torque constants (step 2516). The vector {KF} hasm+2 entries for the m+2 winding-configuration motor. KF(1)=K,KF(j+1)=K/X(j) for j=1 to m, and KF(m+2)=K/N. Next, the method 2500calculates the vector of maximum angular velocities {Max_V}, whereswitching to the next winding-configuration (next KF) occurs (step2518), with Max_V(1)=V, Max_V(j+1)=X(j)*V for j=1 to m, andMax_V(m+2)=NV. In one embodiment, the method 2500 ends at step 2518,whereby the method 2500 for dynamic coil switching is configured oncefor a motor and never changed again.

One example of method 2500 could apply to DVD-ROM and Blu-Ray-ROM diskdrives, both of which read data from the respective optical disk at aconstant linear velocity (CLV). The layout of a DVD-ROM disk 2600 isshown in FIG. 26, where FIG. 26 is a layout diagram of an optical disk2600, with dimensions coming from ECMA-267, entitled 120 mmDVD—Read-Only Disk. The R_outer_physical of the DVD disk, 2612, equals60 mm (120 mm diameter disk) and the R_inner_physical of the DVD disk,2602, equals 7.5 mm (15 mm diameter central hole). Blu-Ray disks havethe same physical inner and outer radii, so that DVD disks and Blu-Raydisks may be played on the same drive. DVD disks have a lead in zonebeginning at 2604 R_inner_lead-in at 22.6 mm. The data begins at 2606R_inner_data, at 24 mm. The data continues to 2608 R_outer_data, at 57.5mm to 58 mm, where the lead-out zone begins. The lead-out zoneterminates at 2610 R_outer_lead-out, at approximately 58.5 mm. Method2500 allows the motor of the disk player to have a torque constant andvoltage constant (Kt=Kv in SI units) commensurate to the outermostactive radius of the disk, 2610 R_outer_lead-out, and the constantlinear velocity (CLV) of the disk.CLV=V*R_outer_lead-out=N*V*R_inner_lead-in. Solving for N for a DVDdisk, we get N=R_outer_lead-out/R_inner_lead-in=58.5/22.6=2.6. N may berounded up to 3 to account for drive, motor, and disk tolerances, givinga practical application of FIG. 8, if m=1 equations are chosen. As thedrive reads data from the inner radius outwards, the disk is slowed downand the algorithm shown in FIG. 8 is utilized from bottom to top. Then,if the disk is dual layer, and additional data is read from the outerradius inwards, the algorithm shown in FIG. 8 is utilized from top tobottom.

However, method 2500 permits an additional embodiment, whereby thealgorithm for dynamic coil switching is itself dynamicallyreprogrammable, thus allowing continual change of multiplicative factorN and/or number of equations m. Method 2500 continues to step 2520 wherea check is made whether to increase the number of equations m, as thatpermits higher performance, or to decrease the number of equations m,for lower performance. If the answer is yes in step 2520, method 2500transfers to step 2522 and the new value of m is obtained, where m isgreater than or equal to one (e.g., m>1), and a flag is set that m hasbeen changed. If the answer is no in step 2520, method 2500 transfers tostep 2524, where a check is made whether to increase multiplicativefactor N for higher maximum motor speed and simultaneously less motortorque, or to reduce multiplicative factor N for lower maximum motorspeed, and simultaneously more motor torque (step 3524). If the answeris yes in step 2524, method 2500 transfers to step 2526 and a new valueof m (where m is greater than m, m>1) is obtained and method 2500proceeds to step 2508 to begin recalculation of the switching algorithm.If the answer is no in step 2524, method 2500 transfers to step 2528where the flag is checked from step 2520. If the answer is yes in step2528, method 2500 proceeds to step 2508 to begin recalculation of theswitching algorithm. If the answer is no in step 2528, the algorithmloops back to step 2520, searching for changes in m and N.

There are numerous applications for dynamically altering control method2500 with additional steps 2520-2528. In one embodiment, the constantangular-velocity of disks in a hard disk drive (HDD) is run at a lowangular-velocity V to save on power expended during low disk SIOs (StartInput/Output Activities). Running at a low angular-velocity V permitsthe storage array of hard disk drives to address workload as the storagearray is not in a sleep mode but it is clearly in a power-saving mode.As the queue of pending SIOs builds, N is dynamically increased toaccommodate the increased workload at the higher angular-velocity of NV.Once the workload becomes quiescent, the value of N is correspondinglyreduced.

Another application for dynamically altering control method 2500involves automobiles, trucks, and motorcycles. Parents may wish todowngrade the performance of the car, if their teenage son or daughterwas taking out the car, and V (an angular-velocity) was equivalent to 15MPH, then the parent or guardian could set N=4, meaning that 4V=60 MPHand the car would be speed limited to 60 MPH (fast enough for anyteenage driver). Additionally, the motor could have a limited number ofwinding-configurations (e.g., switching-states), say m+2=3, meaning thatthe car would take longer to accelerate to the maximum velocity (theautomotive equivalent of a 3-speed transmission) rather than m+2=5 (forexample) which would be reserved for the parents (the automotiveequivalent of a 5-speed transmission). Thus, the parents have tuned downthe performance of the car, given the lack of experience of the driver.The parents would have a password, one allowing the parents to set N=5(allowing the parents to go 75 MPH) and m+2=5 (faster acceleration). Theidea here is not that we merely have a control algorithm for coilswitching “on the fly”, but that algorithm itself can be reconfigured onthe fly (nearly instantaneously), depending upon (a) age of the driver,(b) driving conditions such as weather, and (c) time of day. The carcould go to a lower performance mode for night driving (3N=45 MPH),especially if the weather was bad and road traction was poor due torain, snow, or ice.

The embodiments are going beyond steps 2502-2518 of method 2500 ofsimply dynamically reconfiguring the motor coils. Now the presentinvention is dynamically reconfiguring the algorithm/calculationsitself, which is easily done, given the tridiagonal set of equations hasan easily programmed solution that avoids the complexity of inverting anm-by-m matrix because the tridiagonal [A] matrix is “sparse” (mostlyzeros). Yet another embodiment is that the maximum velocity of a car isdetermined by N*V*wheel radius. N could be fed to the car via wirelesscommunication such as cell phone telepathy, or Bluetooth communication,or GPS-location, and thus, speeding would be impossible in school zones,construction zones, residential streets, etc. Bluetooth is anadvantageous method of communication, given its intentionallyshort-range distance of communication. Essentially, the car wouldreceive the legal value of N via Bluetooth transmitters in the roadway,or via GPS-location, and the maximum velocity of the car is set thatway. N=1 gives 15 MPH for school zones. N=2 or N=3 gives 30 or 45 MPHfor residential or business zones. N=4 (60 MPH) or N=5 (75 MPH) gives“expressway” speeds. Bluetooth or GPS-location gives N to the car, andthe car resets its algorithm in a fraction of a second, to accommodatethe newly allowed speed limit). To allow for varying models of cars, thesame Bluetooth network may send out N based on the model of the car,e.g. Honda_N=2, Toyota_N=3, Ford_N=1, F150_N=4, etc.

In yet another automobile, truck, bus, or motorcycle embodiment, if anaccident occurs, the Bluetooth network lowers the value of N to speedregulate motorists past the accident scene, to prevent massive chainreaction accidents, especially when visibility is limited.

In a medical embodiment, the DC brushless motor turns an Archimedesscrew blood-flow assist of an artificial heart. The DC brushless motorcould be held in place by being part of a medical stent used inarteries. As the oxygen content goes down, as measured by a medicaloxygen sensor, N is dynamically increased to allow more blood flowthrough the lungs. Similarly, N could be increased for dental drillapplications requiring a higher angular-velocity.

FIG. 27 is a block diagram 2700 illustrating an exemplary process formonitoring the angular-velocity of the electric motor. Using the HallEffect Sensor 190, the electric motor's angular-velocity is measured.The microprocessors 192 monitors the angular-velocity of the electricmotor, and upon reaching maximum angular velocities Max_V(j) (shown as171, 161, 151, 141, 121, and 121), the microprocessors dynamicallychanges the effective motor coil length to effect KF(j+1), with coils107, 106, 105, 104, 103, 102, and 101.

FIG. 28 is a block diagram illustrating an exemplary motor used in amedical device 2800. Using a brushless DC motor 2802 with the variousembodiments described herein to dynamically control the calculationfunction and the dynamic switching of the motors 2802 windings, thepropeller 2804 of the motor 2802 is used to assist the blood flow 2806in an aorta 2812 from the heart 2808 for a patient. It should be notedthat the motor 2802 may be any type of motor, including but not limitedto, an electric motor and/or a brushless DC motor. Also, it should benoted that the medical device 2800 may use a variety of motor types,depending upon the particular need of the patient and the specificmedical device 2800. The medical device 2800 may use the motor 2802,which employs the various embodiments described herein, in a variety ofsettings and services to assist the health and welfare of a patient. Forexample, an electric motor 2802 may be used in a medical device, such asan electric wheel chair powered by the motor 2802 while employing thevarious methods described herein, such as in FIG. 2500 which controlsthe calculation function on the fly.

FIG. 29 is a flowchart illustrating an exemplary method 2900 foroptimizing a dynamic reconfiguration-switching of motor windings in anelectric motor. The method 2900 begins (step 2902) by optimizing adynamic reconfiguration-switching of motor windings betweenwinding-configurations for trading off acceleration in favor of highervelocity upon detecting the electric motor is at an optimalangular-velocity for switching to an optimal lower torque constant andvoltage constant (step 2904). The total back electromotive force (BEMF)is prohibited from inhibiting further acceleration to a higherangular-velocity (step 2906). The method 2900 ends (step 2908).

As mentioned above, the embodiments of the present invention may applyto a motor, an electric motor, a brushless DC motor, a tape drivesystem, an electric motor in an electric vehicle or motorcycle, a windturbine, and/or a variety of other types of motors. However, each ofthese various motor types have physical properties and/or hardwareconfigurations that are separate and distinct. For example, an electriccar runs with a constant wheel radius so each of the 4 motors in anelectric car should run at the same angular velocity (except when makinga right or left hand turn). Interestingly enough, if one wheel isspinning at a significantly faster speed than the others, a change inwinding configuration, as described above in each of the Fig's, such asengaging additional motor windings, could be performed to compensate fora loss of traction, hydroplaning, and/or skidding, etc. In such ascenario, it would not be desired to facilitate high speed to a wheellosing fraction, which is counterintuitive to what you'd do in a tapedrive when one tape motor needs to spin faster than the other.

Moreover, the two motors in a tape drive system almost never run at thesame angular velocity. Only at middle-of-tape (MOT) do the two drivemotors run at the same angular velocity. At a beginning of tape (BOT),the supply reel includes all of the tape, and hence has a large radius,and thus, rotates more slowly than the take-up reel (with its smallradius because it has no tape). Hence, as tape is spun off of the supplyreel, the supply reel speeds up, and as tape is added to the takeupreel, the take up reel slows down. A tape drive may have N=3 for anormal data I/O speed. However, N=6 (or some other value) may be usedfor high speed searching or rewinding. Hence, the tape drive is a goodexample of where multiple values of N may be used.

In addition, the following illustrates other differences in the physicalproperties of the motor types for applying the embodiments of thepresent invention. 1) A hard disk drive or an optical disk drive has 1(and only 1) drive motor and the drive motor spins the single stack ofdisks comprising one or more disks. 2) A tape drive has only 2 motors,one for the supply reel and one for the takeup reel. It should be notedthat belts and pulleys have been used to allow only one motor, however,a tape drive, having 2 motors, only run at the same angular velocity atthe special case of MOT (middle of tape). The supply reel alwaysincreases in angular velocity as it sheds tape and the takeup reelalways decreases in angular velocity as it gains tape (delta angularvelocities of the opposite sign). A high-speed seek may require a higher“N” than normal data I/O. 3) An electric car typically has a minimum of3 motors (tricycle approach) or 4 motors (typical car), but a motorcyclemay have 2 motors. Typically, all motors run at the same angularvelocity, except when a turn is made (as the outer wheels in a turn haveto spin faster than the inner wheels as they have further to go. Anerror condition exists (such as hydroplaning, skidding, etc.) when onewheel turns significantly faster than the others, and rather thanenabling a higher angular velocity by shedding windings it is possibleto add windings to slow down the motor and get more traction, using theembodiments described above in the Fig.'s, but in a torque-control moderather than a velocity-enabling mode.

As will be appreciated by one skilled in the art, aspects of the presentinvention may be embodied as a system, method or computer programproduct. Accordingly, aspects of the present invention may take the formof an entirely hardware embodiment, an entirely software embodiment(including firmware, resident software, micro-code, etc.) or anembodiment combining software and hardware aspects that may allgenerally be referred to herein as a “circuit,” “module” or “system.”Furthermore, aspects of the present invention may take the form of acomputer program product embodied in one or more computer readablemedium(s) having computer readable program code embodied thereon.

Any combination of one or more computer readable medium(s) may beutilized. The computer readable medium may be a computer readable signalmedium or a computer readable storage medium. A computer readablestorage medium may be, for example, but not limited to, an electronic,magnetic, optical, electromagnetic, infrared, or semiconductor system,apparatus, or device, or any suitable combination of the foregoing. Morespecific examples (a non-exhaustive list) of the computer readablestorage medium would include the following: an electrical connectionhaving one or more wires, a portable computer diskette, a hard disk, arandom access memory (RAM), a read-only memory (ROM), an erasableprogrammable read-only memory (EPROM or Flash memory), an optical fiber,a portable compact disc read-only memory (CD-ROM), an optical storagedevice, a magnetic storage device, or any suitable combination of theforegoing. In the context of this document, a computer readable storagemedium may be any tangible medium that can contain or store a programfor use by or in connection with an instruction execution system,apparatus, or device.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wired, optical fiber cable, RF, etc., or any suitable combination of theforegoing. Computer program code for carrying out operations for aspectsof the present invention may be written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Java, Smalltalk, C++ or the like and conventionalprocedural programming languages, such as the “C” programming languageor similar programming languages. The program code may execute entirelyon the user's computer, partly on the user's computer, as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server. In the latterscenario, the remote computer may be connected to the user's computerthrough any type of network, including a local area network (LAN) or awide area network (WAN), or the connection may be made to an externalcomputer (for example, through the Internet using an Internet ServiceProvider).

Aspects of the present invention have been described above withreference to flowchart illustrations and/or block diagrams of methods,apparatus (systems) and computer program products according toembodiments of the invention. It will be understood that each block ofthe flowchart illustrations and/or block diagrams, and combinations ofblocks in the flowchart illustrations and/or block diagrams can beimplemented by computer program instructions. These computer programinstructions may be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructionswhich execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

These computer program instructions may also be stored in a computerreadable medium that can direct a computer, programmable data processingapparatus, or other device to function in a particular manner, such thatthe instructions stored in the computer readable medium produce anarticle of manufacture including instructions which implement thefunction/act specified in the flowchart and/or block diagram block orblocks. The computer program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or device tocause a series of operational steps to be performed on the computer,other programmable apparatus or device, to produce a computerimplemented process such that the instructions which execute on thecomputer or other programmable apparatus provide processes forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks.

The flowchart and block diagrams in the above figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof code, which comprises one or more executable instructions forimplementing the specified logical function(s). It should also be notedthat, in some alternative implementations, the functions noted in theblock may occur out of the order noted in the figures. For example, twoblocks shown in succession may, in fact, be executed substantiallyconcurrently, or the blocks may sometimes be executed in the reverseorder, depending upon the functionality involved. It will also be notedthat each block of the block diagrams and/or flowchart illustration, andcombinations of blocks in the block diagrams and/or flowchartillustration can be implemented by special purpose hardware-basedsystems that perform the specified functions or acts, or combinations ofspecial purpose hardware and computer instructions.

While one or more embodiments of the present invention have beenillustrated in detail, the skilled artisan will appreciate thatmodifications and adaptations to those embodiments may be made withoutdeparting from the scope of the present invention as set forth in thefollowing claims.

1. A method for switching motor windings within an electric motor usinga processor device, the method comprising: optimizing a dynamicreconfiguration-switching of a plurality of motor windings between aplurality of winding-configurations for trading off acceleration infavor of higher velocity upon detecting the electric motor is at anoptimal angular-velocity for switching to an optimal lower torqueconstant and voltage constant thereby preventing a total backelectromotive force (BEMF) from inhibiting further acceleration to ahigher angular-velocity.
 2. The method of claim 1, further includingperforming the dynamic reconfiguration-switching between the pluralityof motor windings using one of a wye connection and a delta connectionsystem for the electric motor.
 3. The method of claim 1, furtherincluding performing the dynamic reconfiguration-switching between theplurality of winding-configurations in a successive order when avelocity sensor detects the electric motor is at the optimalangular-velocity for switching the torque constant and the voltageconstant.
 4. The method of claim 1, further including trading offangular-acceleration for a greater angular-velocity as speed of theelectric motor is increased whereby the torque constant, the voltageconstant, and the angular-acceleration correspondingly decrease.
 5. Themethod of claim 1, further including performing the dynamicreconfiguration-switching using a higher angular-acceleration at a lowerangular-velocity upon slowing down the electric motor.
 6. The method ofclaim 1, further including using m simultaneous-linear equations, havingm number of unknown variables, for determining a plurality of voltageconstants, the optimal time, and the optimal angular-velocity forperforming the dynamic reconfiguration-switching between each of theplurality of winding-configurations, wherein a number of the msimultaneous-linear equations increases, starting with one equation fora three-winding-configuration of the electric motor and adding anadditional simultaneous equation for each additional one of theplurality of winding-configurations, and a number of the plurality ofvoltage constants equal to the correlating one of the plurality ofwinding-configurations.
 7. The method of claim 6, further includingorganizing the m simultaneous-linear equations, having the m number ofunknown variables, into a tridiagonal system of linear equations, acoefficient matrix of the tridiagonal system of linear equations having2's along an entire main diagonal of the coefficient matrix, andnegative 1's along diagonals immediately adjacent to the main diagonal,and all other entries of the coefficient matrix being zero, and aright-hand-side vector comprising a first entry of 1, a last entry of N,denoting N times a maximum allowable angular-velocity V of the electricmotor at a one hundred percent torque constant, and all other entries ofthe right-hand-side vector being zero.
 8. The method of claim 7, furtherincluding, for a WCth-winding-configuration of the electric motor,performs each of: increasing to an angular-velocity N times a capabilityof the electric motor with the one hundred percent torque constant andthe voltage constant, calculating an optional time for performing thedynamic reconfiguration-switching by first finding a total time foracceleration to an angular-velocity of NV and differentiating the totaltime with respect to vector of unknowns X and then setting each firstderivatives to zero to find an optimal value of the vector of unknowns Xto minimize the total time, and taking a second derivative of the totaltime with respect to the vector of unknowns X, wherein a positive-secondderivative indicates a value of the vector of unknowns X that minimizesand optimizes the total time for performing the dynamicreconfiguration-switching, where X is an unknown used for subdividingthe one hundred percent torque constant of the electric motor intosmaller units of the torque constant which allows the electric motor togo faster than V angular-velocity and up to the angular-velocity of NVwhere V is a maximum angular-velocity achieved with the one hundredpercent torque constant, and N is an arbitrary value greater than 1,where WC is the a number of possible winding-configurations of theelectric motor.
 9. The method of claim 1, further including increasing atotal angular velocity and decreasing the torque constant, the voltageconstant, and angular-acceleration as a number of the plurality of motorwindings of the electric motor increases.
 10. The method of claim 1,further including activating a plurality of switches to connect theplurality of motor windings in a parallel configuration to reduce thetotal back-EMF from the plurality of motor windings and allow forgreater angular-velocity of the electric motor.
 11. The method of claim1, further including activating a plurality of switches to bypass anelectrical connection to at least one of the plurality of motor windingsto provide a minimum back-EMF and allow for greater angular-velocity ofthe electric motor.
 12. The method of claim 1, further includingactivating a plurality of switches to connect the plurality of motorwindings in a serial configuration to maximize torque on the electricmotor.
 13. The method of claim 1, further including selectivelyactivating the plurality of motor windings for each electrical phase ofthe electric motor to provide for multiple velocities.
 14. The method ofclaim 1, further including activating a plurality of switches todisconnect the plurality of motor windings in a serial configuration tomaximize torque on the electric motor.
 15. The method of claim 1,wherein the dynamic reconfiguration-switching occurs between each of theplurality of winding-configurations at an optimal time for allowing adynamic trade-off between the angular-velocity and angular-acceleration.16. A system for switching motor windings within an electric motor, thesystem comprising: an electric motor having a plurality of motorwindings, a plurality of switches for controlling a current through theplurality of motor windings of the electric motor, and a processordevice in communication with the plurality of switches, wherein theprocessor device: optimizes a dynamic reconfiguration-switching ofplurality of motor windings between a plurality ofwinding-configurations for trading off acceleration in favor of highervelocity upon detecting the electric motor is at an optimalangular-velocity for switching to an optimal lower torque constant andvoltage constant thereby preventing a total back electromotive force(BEMF) from inhibiting further acceleration to a higherangular-velocity.
 17. The system of claim 16, wherein the processordevice performs the dynamic reconfiguration-switching between theplurality of motor windings using one of a wye connection and a deltaconnection system for the electric motor.
 18. The system of claim 16,wherein the processor device performs the dynamicreconfiguration-switching between the plurality ofwinding-configurations in a successive order when a velocity sensordetects the electric motor is at the optimal angular-velocity forswitching the torque constant and the voltage constant.
 19. The systemof claim 16, wherein the processor device trades offangular-acceleration for a greater angular-velocity as speed of theelectric motor is increased whereby the torque constant, the voltageconstant, and the angular-acceleration correspondingly decrease.
 20. Thesystem of claim 16, wherein the processor device performs the dynamicreconfiguration-switching using a higher angular-acceleration at a lowerangular-velocity upon slowing down the electric motor.
 21. The system ofclaim 16, wherein the processor device uses m simultaneous-linearequations, having m number of unknown variables, for determining aplurality of voltage constants, the optimal time, and the optimalangular-velocity for performing the dynamic reconfiguration-switchingbetween each of the plurality of winding-configurations, wherein anumber of the m simultaneous-linear equations increases, starting withone equation for a three-winding-configuration of the electric motor andadding an additional simultaneous equation for each additional one ofthe plurality of winding-configurations, and a number of the pluralityof voltage constants equal to the correlating one of the plurality ofwinding-configurations.
 22. The system of claim 21, wherein theprocessor device organizes the m simultaneous-linear equations, havingthe m number of unknown variables, into a tridiagonal system of linearequations, a coefficient matrix of the tridiagonal system of linearequations having 2's along an entire main diagonal of the coefficientmatrix, and negative 1's along diagonals immediately adjacent to themain diagonal, and all other entries of the coefficient matrix beingzero, and a right-hand-side vector comprising a first entry of 1, a lastentry of N, denoting N times a maximum allowable angular-velocity V ofthe motor at a one hundred percent torque constant, and all otherentries of the right-hand-side vector being zero.
 23. The system ofclaim 22, wherein the processor device, for a WCth-winding-configurationof the electric motor, performs each of: increasing to anangular-velocity N times a capability of the electric motor with the onehundred percent torque constant and the voltage constant, calculating anoptional time for performing the dynamic reconfiguration-switching byfirst finding a total time for acceleration to an angular-velocity of NVand differentiating the total time with respect to vector of unknowns Xand then setting each first derivatives to zero to find an optimal valueof the vector of unknowns X to minimize the total time, and taking asecond derivative of the total time with respect to the vector ofunknowns X, wherein a positive-second derivative indicates a value ofthe vector of unknowns X that minimizes and optimizes the total time forperforming the dynamic reconfiguration-switching, where X is an unknownused for subdividing the one hundred percent torque constant of theelectric motor into smaller units of the torque constant which allowsthe electric motor to go faster than V angular-velocity and up to theangular-velocity of NV where V is a maximum angular-velocity achievedwith the one hundred percent torque constant, and N is an arbitraryvalue greater than 1, where WC is the a number of possiblewinding-configurations of the electric motor.
 24. The system of claim23, wherein the processor device increases a total angular velocity anddecreasing the torque constant, the voltage constant, andangular-acceleration as a number of the plurality of motor windings ofthe electric motor increases.
 25. The system of claim 16, wherein theprocessor device activates the plurality of switches to connect theplurality of motor windings in a parallel configuration to reduce thetotal back-EMF from the plurality of motor windings and allow forgreater angular-velocity of the electric motor.
 26. The system of claim16, wherein the processor device activates the plurality of switches tobypass an electrical connection to at least one of the plurality ofmotor windings to provide a minimum back-EMF and allow for greaterangular-velocity of the electric motor.
 27. The system of claim 16,wherein the processor device activates the plurality of switches toconnect the plurality of motor windings in a serial configuration tomaximize torque on the electric motor.
 28. The system of claim 16,wherein the processor device selectively activates the plurality ofmotor windings for each electrical phase of the electric motor toprovide for multiple velocities.
 29. The system of claim 16, wherein theprocessor device activates the plurality of switches to disconnect theplurality of motor windings in a serial configuration to maximize torqueon the electric motor.
 30. The system of claim 16, wherein the dynamicreconfiguration-switching occurs between each of the plurality ofwinding-configurations at an optimal time for allowing a dynamictrade-off between the angular-velocity and angular-acceleration.
 31. Acomputer program product for switching motor windings within an electricmotor using a processor device, the computer program product comprisinga non-transitory computer-readable storage medium havingcomputer-readable program code portions stored therein, thecomputer-readable program code portions comprising: a first executableportion that optimizes a dynamic reconfiguration-switching of pluralityof motor windings between a plurality of winding-configurations fortrading off acceleration in favor of higher velocity upon detecting theelectric motor is at an optimal angular-velocity for switching to anoptimal lower torque constant and voltage constant thereby preventing atotal back electromotive force (BEMF) from inhibiting furtheracceleration to a higher angular-velocity.
 32. The computer programproduct of claim 31, further including a second executable portion thatperforms the dynamic reconfiguration-switching between the plurality ofmotor windings using one of a wye connection and a delta connectionsystem for the electric motor.
 33. The computer program product of claim31, further including a second executable portion that performs thedynamic reconfiguration-switching between the plurality ofwinding-configurations in a successive order when a velocity sensordetects the electric motor is at the optimal angular-velocity forswitching the torque constant and the voltage constant.
 34. The computerprogram product of claim 31, further including a second executableportion that trades off angular-acceleration for a greaterangular-velocity as speed of the electric motor is increased whereby thetorque constant, the voltage constant, and the angular-accelerationcorrespondingly decrease.
 35. The computer program product of claim 31,further including a second executable portion that performs the dynamicreconfiguration-switching using a higher angular-acceleration at a lowerangular-velocity upon slowing down the electric motor.
 36. The computerprogram product of claim 31, further including a second executableportion that uses m simultaneous-linear equations, having m number ofunknown variables, for determining a plurality of voltage constants, theoptimal time, and the optimal angular-velocity for performing thedynamic reconfiguration-switching between each of the plurality ofwinding-configurations, wherein a number of the m simultaneous-linearequations increases, starting with one equation for athree-winding-configuration of the electric motor and adding anadditional simultaneous equation for each additional one of theplurality of winding-configurations, and a number of the plurality ofvoltage constants equal to the correlating one of the plurality ofwinding-configurations.
 37. The computer program product of claim 36,further including a third executable portion that organizes the msimultaneous-linear equations, having the m number of unknown variables,into a tridiagonal system of linear equations, a coefficient matrix ofthe tridiagonal system of linear equations having 2's along an entiremain diagonal of the coefficient matrix, and negative 1's alongdiagonals immediately adjacent to the main diagonal, and all otherentries of the coefficient matrix being zero, and a right-hand-sidevector comprising a first entry of 1, a last entry of N, denoting Ntimes a maximum allowable angular-velocity of the motor at a one hundredpercent torque constant, and all other entries of the right-hand-sidevector being zero.
 38. The computer program product of claim 37, furtherincluding a fourth executable portion that, for aWCth-winding-configuration of the electric motor, performs each of:increasing to an angular-velocity N times a capability of the electricmotor with the one hundred percent torque constant and the voltageconstant, calculating an optional time for performing the dynamicreconfiguration-switching by first finding a total time for accelerationto an angular-velocity of NV and differentiating the total time withrespect to vector of unknowns X and then setting each first derivativesto zero to find an optimal value of the vector of unknowns X to minimizethe total time, and taking a second derivative of the total time withrespect to the vector of unknowns X, wherein a positive-secondderivative indicates a value of the vector of unknowns X that minimizesand optimizes the total time for performing the dynamicreconfiguration-switching, where X is an unknown used for subdividingthe one hundred percent torque constant of the electric motor intosmaller units of the torque constant which allows the electric motor togo faster than V angular-velocity and up to the angular-velocity of NVwhere V is a maximum angular-velocity achieved with the one hundredpercent torque constant, and N is an arbitrary value greater than 1,where WC is the a number of possible winding-configurations of theelectric motor.
 39. The computer program product of claim 38, furtherincluding a fifth executable portion that increases a total angularvelocity and decreasing the torque constant, the voltage constant, andangular-acceleration as a number of the plurality of motor windings ofthe electric motor increases.
 40. The computer program product of claim31, further including a second executable portion that activates theplurality of switches to connect the plurality of motor windings in aparallel configuration to reduce the total back-EMF from the pluralityof motor windings and allow for greater angular-velocity of the electricmotor.
 41. The computer program product of claim 31, further including asecond executable portion that activates the plurality of switches tobypass an electrical connection to at least one of the plurality ofmotor windings to provide a minimum back-EMF and allow for greaterangular-velocity of the electric motor.
 42. The computer program productof claim 31, further including a second executable portion thatactivates the plurality of switches to connect the plurality of motorwindings in a serial configuration to maximize torque on the electricmotor.
 43. The computer program product of claim 31, further including asecond executable portion that selectively activates the plurality ofmotor windings for each electrical phase of the electric motor toprovide for multiple velocities.
 44. The computer program product ofclaim 31, further including a second executable portion that activatesthe plurality of switches to disconnect the plurality of motor windingsin a serial configuration to maximize torque on the electric motor. 45.The computer program product of claim 31, wherein the dynamicreconfiguration-switching occurs between each of the plurality ofwinding-configurations at an optimal time for allowing a dynamictrade-off between the angular-velocity and angular-acceleration.